The Symplectic Sum Formula for Gromov-Witten Invariants

Annals of Mathematics, 159 (2004), 935–1025

The symplectic sum formula for Gromov-Witten invariants

By Eleny-Nicoleta Ionel and Thomas H. Parker*

Abstract In the symplectic category there is a ‘connect sum’ operation that glues symplectic manifolds by identifying neighborhoods of embedded codimension two submanifolds. This paper establishes a formula for the Gromov-Witten invariants of a symplectic sum Z = X#Y in terms of the relative GW invariants of X and Y . Several applications to enumerative geometry are given.

Gromov-Witten invariants are counts of holomorphic maps into symplectic manifolds. To define them on a symplectic manifold (X, ω) one introduces an almost complex structure J compatible with the symplectic form ω and forms the moduli space of J-holomorphic maps from complex curves into X and the compactified moduli space, called the space of stable maps. One then imposes constraints on the stable maps, requiring the domain to have a certain form and the image to pass through fixed homology cycles in X. When the correct number of constraints is imposed there are only finitely many maps satisfying the constraints; the (oriented) count of these is the corresponding GW invariant. For complex algebraic manifolds these symplectic invariants can also be defined by algebraic geometry, and in important cases the invariants are the same as the curve counts that are the subject of classical enumerative algebraic geometry. In the past decade the foundations for this theory were laid and the invariants were used to solve several long-outstanding problems. The focus now is on finding effective ways of computing the invariants. One useful technique is the method of ‘splitting the domain’, in which one localizes the invariant to the set of maps whose domain curves have two irreducible components with the constraints distributed between them. This produces recursion relations relating the desired GW invariant to invariants with lower degree or genus. This paper establishes a general formula describing the behavior of GW invariants under the analogous operation of ‘splitting the target’. Because we

*The research of both authors was partially supported by the NSF. The first author was also supported by a Sloan Research Fellowship. 936 ELENY-NICOLETA IONEL AND THOMAS H. PARKER work in the context of symplectic manifolds the natural splitting of the target is the one associated with the symplectic cut operation and its inverse, the symplectic sum. The symplectic sum is defined by gluing along codimension two submanifolds. Specifically, let X be a symplectic 2n-manifold with a symplectic (2n−2)- submanifold V . Given a similar pair (Y,V ) with a symplectic identification between the two copies of V and a complex anti-linear isomorphism between the normal bundles NX V and NY V of V in X and in Y , we can form the symplectic sum Z = X#V Y . Our main theorem is a ‘Symplectic Sum Formula’ which expresses the GW invariants of the sum Z in terms of the relative GW invariants of (X, V ) and (Y,V ) introduced in [IP4]. The symplectic sum is perhaps more naturally seen not as a single manifold but as a family depending on a ‘squeezing parameter’. In Section 2 we construct a family Z → D over the disk whose fibers Zλ are smooth and symplectic for λ = 0 and whose central fiber Z0 is the singular manifold X ∪V Y .Ina neighborhood of V , the total space Z is NX V ⊕ NY V and the fiber Zλ is defined by the equation xy = λ where x and y are coordinates in the normal ∼ ∗ bundles NX V and NY V = (NX V ) . The fibration Z → D extends away from V as the disjoint union of X × D and Y × D, and the entire fibration Z can be given an almost Kähler structure. The smooth fibers Zλ, depicted in Figure 1, are symplectically isotopic to one another; each is a model of the symplectic sum. The overall strategy for proving the symplectic sum formula is to relate the holomorphic maps into Z0, which are simply maps into X and Y which match along V , with the holomorphic maps into Zλ for λ close to zero. This strategy involves two parts: limits and gluing. For the limiting process we consider sequences of stable maps into the family Zλ of symplectic sums as the ‘neck size’ λ → 0. In particular, these are stable maps into a compact region of the almost Kähler manifold Z, so that the compactness theorem for stable maps applies, giving limit maps into the singular manifold Z0 obtained by identifying X and Y along V . Along the way several things become apparent. First, the limit maps are holomorphic only if the almost complex structures on X and Y match along V . To ensure this we impose the “V -compatibility” condition (1.10) on the almost complex structure. But there is a price to pay for that specialization. In the symplectic theory of Gromov-Witten invariants we are free to perturb (J, ν) without changing the invariant; this freedom can be used to ensure that intersections are transverse. After imposing the V -compatibility condition, we can no longer perturb (J, ν) along V at will, and hence we cannot assume that the limit curves are transverse to V . In fact, the images of the components of the limit maps meet V at points with well-defined multiplicities and, worse, some components may be mapped entirely into V . THE SYMPLECTIC SUM FORMULA 937

To count stable maps into Z0 we look first on the X side and ignore the maps which have marked points, double points, or whole components mapped into V . The remaining “V -regular” maps form a moduli space which is the MV union of components s (X) labeled by the multiplicities s =(s1,...,s)of MV the intersection points with V . We showed in [IP4] how these spaces s (X) can be compactified and used to define relative Gromov-Witten invariants V GWX . The definitions are briefly reviewed in Section 1.

Figure 1. Limiting curves in Zλ = X#λY as λ → 0.

Second, as Figure 1 illustrates, connected curves in Zλ can limit to curves whose restrictions to X and Y are not connected. For that reason the GW invariant, which counts stable curves from a connected domain, is not the appropriate invariant for expressing a sum formula. Instead one should work with the ‘Gromov-Taubes’ invariant GT, which counts stable maps from domains that need not be connected. Thus we seek a formula of the general form V ∗ V (0.1) GTX GTY =GTZ where ∗ is some operation that adds up the ways curves on the X and Y sides match and are identified with curves in Zλ. That necessarily involves keeping track of the multiplicities s and the homology classes. It also involves accounting for the limit maps with nontrivial components in V ; such curves are not counted by the relative invariant and hence do not contribute to the left side of (0.1). We postpone this issue by first analyzing limits of curves which are δ-flat in the sense of Definition 3.1. A more precise analysis reveals a third complication: the squeezing process is not injective. In Section 5 we again consider a sequence of stable maps fn into Zλ as λ → 0, this time focusing on their behavior near V , where the fn do not uniformly converge. We form renormalized maps fˆn and prove that both the domains and the images of the renormalized maps converge. The images converge nicely according to the leading order term of their Taylor expansions, but the domains converge only after we fix certain roots of unity. These roots of unity are apparent as soon as one writes down formulas. Each stable map f : C → Z0 decomposes into a pair of maps f1 : C1 → X and f2 : C2 → Y which agree at the nodes of C = C1 ∪ C2. For a specific example, suppose that f is such a map that intersects V at a single point p 938 ELENY-NICOLETA IONEL AND THOMAS H. PARKER with multiplicity three. Then we can choose local coordinates z on C1 and w on C2 centered at the node, and coordinates x on X and y on Y so that f1 3 3 and f2 have expansions x(z)=az + ··· and y(w)=bw + ···. To find maps into Zλ near f, we smooth the domain C to the curve Cµ given locally near the node by zw = µ and require that the image of the smoothed map lie in Zλ, which is locally the locus of xy = λ. In fact, the leading terms in the formulas for f1 and f2 define a map F : Cµ → Zλ whenever

λ = xy = az3 · bw3 = ab (zw)3 = ab µ3 and conversely every family of smooth maps which limit to f satisfies this equation in the limit (cf. Lemma 5.3). Thus λ determines the domain Cµ up to a cube root of unity. Consequently, this particular f is, at least a priori, close to three smooth maps into Zλ — a ‘cluster’ of order three. Other maps f into Z0 have larger associated clusters (the order of the cluster is the product of the multiplicities with which f intersects V ). The maps within a cluster have the same leading order formula but have different smoothings of the domain. As λ → 0 the cluster coalesces, limiting to the single map f. This clustering phenomenon greatly complicates the analysis. To distinguish the curves within each cluster and make the analysis uniform in λ as λ → 0, it is necessary to use ‘rescaled’ norms and distances which magnify distances as the clusters form. With the right choice of norms, the distances between the maps within a cluster are bounded away from zero as λ → 0 and become the fiber of a covering of the space of limit maps. Sections 4–6 introduce the required norms, first on the space of curves, then on the space of maps. For maps we use a Sobolev norm weighted in the directions perpendicular to V ; the weights are chosen so the norm dominates the C0 distance between the renormalized maps fˆ. On the space of curves we require a stronger metric than the usual complete metrics on Mg,n. In Section 4 we define a complete metric on Mg,n \N where N is the set of all nodal curves. In this metric the distance between two sequences that approach N from different directions (corresponding to the roots of unity mentioned above) is bounded away from zero; thus this metric separates the domain curves of maps within a cluster. This construction also leads to a compactification of Mg,n \N in which the stratum N of -nodal curves is replaced by a bundle over N whose fiber is the real torus T . The limit process is reversed by constructing a space of approximately holomorphic maps and showing it is diffeomorphic to the space of stable maps into Zλ. The space of approximate maps is described in Section 6, first intrinsically, then as a subset Models(Zλ) of the space of maps. For each s and λ it is a covering of the space Ms(Z0) of the δ-flat maps into Z0 that meet THE SYMPLECTIC SUM FORMULA 939

V at points with multiplicities s. The fibers of this covering are the clusters described above; they are distinct maps into Zλ which converge to the same limit as λ → 0. From there the analysis follows a standard technique that goes back to Taubes and Donaldson: correct the approximate maps to true holomorphic maps by constructing a partial right inverse to the linearization D and applying a fixed point theorem. This involves (a) showing that the operator D∗D is uniformly invertible as λ → 0, and (b) proving a priori that every solution is close to an approximate solution, close enough to be in the domain of the fixed point theorem. Proposition 9.3 shows that (b) follows from the renormalization analysis of Section 4. But the eigenvalue estimate (a) proves to be surprisingly delicate and seems to succeed only with a very specific choice of norms. The difficulty, of course, is that Zλ becomes singular along V as λ → 0. However, for small λ the bisectional curvature in the neck region is negative; a Bochner formula then shows that eigenfunctions with small eigenvalue cannot be concentrating in the neck. One can then reason that since the cokernel of D vanishes on Z0 for generic V -compatible (J, ν) it should also vanish on Zλ for small λ. We make that reasoning rigorous by introducing exponential weight functions into the norms, thereby making the linearizations Dλ a continuous family of Fredholm maps. That in turn necessitates further work on the Bochner formula, bounding the additional term that arises from the derivative of the weight functions. These estimates are carried out in Section 8. The upshot of the analysis is a diffeomorphism between the approximate moduli space and the true moduli spaces ∼ = Models(Zλ)s −→ M s(Zλ) which intertwines with the attaching map of the domains and the evaluation map into the target (Theorem 10.1). We then pass to homology, comparing and keeping track of the homology classes of the maps, the domains, and the constraints. This involves several difficulties, all ultimately due to the fact that H∗(Zλ) is different from both H∗(Z0) and H∗(X) ⊕ H∗(Y ). This is sorted out in Section 10, where we define the convolution operation and prove a first Symplectic Sum Theorem: a formula like (0.1) holds when all stable maps are δ-flat. In Sections 11 and 12 we remove the δ-flatness assumption by partitioning the neck into a large number of segments and using the pigeon-hole principle N as in Wieczorek [W]. For that we construct spaces Zλ (µ1,...,µ2N+1), each symplectically isotopic to Zλ.As(µ1,...,µ2N+1) → 0 these degenerate to the singular space obtained by connecting X to Y through a series of 2N copies of the rational ruled manifold PV obtained by adding an infinity section to the normal bundle to V . An energy bound shows that for large N each map into Zλ(µ1,...,µ2N+1) must be flat in most necks. Squeezing some or all of the flat 940 ELENY-NICOLETA IONEL AND THOMAS H. PARKER necks decomposes the curves in Zλ into curves in X joined to curves in Y by a chain of curves in intermediate spaces PV . The limit maps are then δ-flat, so that formula (0.1) applies to each. This process counts each stable map many times (there are many choices of where to squeeze) and in fact gives an open cover of the moduli space. Working through the combinatorics and inverting a power series, we show that the total contribution of the entire neck region between X and Y is given by a certain GT invariant of PV — the S-matrix of Definition 11.3.

XYF F

Figure 2. Zλ(µ, µ, µ) for |µ||λ|

The S-matrix keeps track of how the genus, homology class, and intersection points with V change as the images of stable maps pass through the middle region of Figure 2. Observing this back in the model of Figure 1, one sees these quantities changing abruptly; as the maps pass through the neck, they are “scattered” by the neck. The scattering occurs when some of the stable maps contributing to the GT invariant of Zλ have components that lie entirely in V in the limit as λ → 0. Those maps are not V -regular, so are not counted in the relative invariants of X or Y . But by moving to the spaces of Figure 2 this complication can be analyzed and related to the relative invariants of the ruled manifold PV . The S-matrix is the ﬁnal subtlety. With it in hand, we can at last state our main result. Symplectic Sum Theorem. Let Z be the symplectic sum of (X, V ) and ∗ (Y,V ) and suppose that α ∈ T (Z) splits as (αX ,αY ) as in Deﬁnition 10.5. Then the GT invariant of Z is given in terms of the relative invariants of X and Y by V ∗ ∗ V (0.2) GTZ (α)=GTX (αX ) SV GTY (αY ) where ∗ is the convolution operation (10.6) and SV is the S-matrix (11.3). A detailed statement of this theorem is given in Section 12 and its extension to general constraints α is discussed in Section 13. We actually state and prove (0.2) as a formula for the relative invariants of Z in terms of the relative invariants of X and Y (Theorem 12.3). In that more general form the formula can be iterated. THE SYMPLECTIC SUM FORMULA 941

Of course, (0.2) is of limited use unless we can compute the relative invariants of X and Y and the associated S-matrix. That turns out to be perfectly feasible, at least for simple spaces. In Section 14 we build a collection of two and four dimensional spaces whose relative GT invariants we can compute. We also prove that the S-matrix is the identity in several cases of particular interest. The last section presents applications. The examples of Section 14 are used as building blocks to give short proofs of three recent results in enumerative geometry: (a) the Caporaso-Harris formula for the number of nodal curves in P2 [CH], (b) the formula for the Hurwitz numbers counting branched covers of P1 ([GJV] [LZZ]), and (c) the “quasimodular form” expression for the rational enumerative invariants of the rational elliptic surface ([BL]). In hindsight, our proofs of (a) and (b) are essentially the same as those in the literature; using the symplectic sum formula makes the proof considerably shorter and more transparent, but the key ideas are the same. Our proof of (c), however, is completely different from that of Bryan and Leung in [BL]. It is worth outlining here. The rational elliptic surface E fibers over P1 with a section s and fiber f. For each d ≥ 0 consider the invariant GWd which counts the number of connected rational stable maps in the class s + df . Bryan and Leung showed that d the generating series F0(t)= d GWd t is 12 1 (0.3) F (t)= . 0 1 − td d

This formula is related to the work of Yau-Zaslow [YZ] and is one of the sim- plest instances of some general conjectures concerning counts of nodal curves in complex surfaces — see [Go]. While the intriguing form (0.3) appears in [BL] for purely combinatorial reasons, it arises in our proof because of a connection with elliptic curves. In fact, our proof begins by relating F0 to a similar series H which counts elliptic curves in E. We then regard E as the ﬁber sum E#(T 2 × S2) and apply the symplectic sum formula. The relevant relative invariant on the T 2 × S2 side is easily seen to be the generating function G(t) for the number of degree d coverings of the torus T 2 by the torus. The symplectic sum formula reduces to a diﬀerential equation relating F0(t) with G(t), and integration yields the quasimodular form (0.3). The details, given in Section 15.3, are rather formal; the needed geometric input is mostly contained in the symplectic sum formula. All three of the applications in Section 15 use the idea of ‘splitting the target’ mentioned at the beginning of this introduction. Moreover, all three follow from rather simple cases of the Symplectic Sum Theorem — cases where the S-matrix is the identity and where at least one of the relative invariants 942 ELENY-NICOLETA IONEL AND THOMAS H. PARKER in (0.2) is readily computed by elementary methods. The full strength of the symplectic sum theorem has not yet been used. This paper is a sequel to [IP4]; together with [IP4] it gives a complete detailed exposition of the results announced in [IP3]. Further applications have already appeared in [IP2] and [I]. A. M. Li and Y. Ruan also have a sum formula [LR]. Y. Eliashberg, A. Givental, and H. Hofer are developing a general theory for invariants of symplectic manifolds glued along contact boundaries [EGH].

Contents

1. GW and GT invariants 2. Symplectic sums 3. Degenerations of symplectic sums 4. The space of curves 5. Renormalization at the nodes 6. The space of approximate maps 7. Linearizations 8. The eigenvalue estimate 9. The gluing diﬀeomorphism 10. Convolutions and the sum formula for ﬂat maps 11. The space PV and the S-matrix 12. The general sum formula 13. Constraints passing through the neck 14. Relative GW invariants in simple cases 15. Applications of the sum formula Appendix: Expansions of relative GT invariants

1. GW and GT invariants

For stable maps and their associated invariants we will use the deﬁnitions and notation of [IP4], which build on those of Ruan-Tian [RT1] and [RT2]. Thus we work in the context of a closed symplectic manifold (X, ω) with an almost complex structure J and Riemannian metric g which are compatible in the sense that

(1.1) g(v, w)=ω(v, Jw) ∀v, w ∈ TX.

In summary, the definitions are as follows. A bubble domain B is a finite connected union of smooth oriented 2-manifolds Bi joined at nodes together with n marked points pi, none of which are nodes. Collapsing the unstable components to points gives a connected domain st(B). Let U g,n → Mg,n be the universal curve over the Deligne-Mumford space of genus g curves with n THE SYMPLECTIC SUM FORMULA 943 marked points. We can put a complex structure j on B, well-defined up to diffeomorphism, by specifying an orientation-preserving map φ0 :st(B) → U g,n which is a diffeomorphism onto a fiber of U g,n and taking the standard complex structure on the unstable (rational) components. We will often write C for the curve (B,j,p1,...,pn) and use the notation (f,C)orf : C → X instead of (f,φ). A(J, ν)-holomorphic map from B is a map (f,φ):B → X × U g,n where φ = φ0◦st and whose restriction to each irreducible component Bi of B satisfies ∗ (1.2) ∂¯J f =(f,φ) ν. ¯ 1 U ⊂ PN Here ∂J f means 2 (df + Jf df j φ) and, after an embedding g,n is fixed, ν ∗ PN ∗ × PN is a section of the bundle Hom(π1T ,π2TX) over X satisfying Jν= −νJPN . Alternatively, we can define an almost complex structure Jˆ on X×U g,n by Jˆ(u, v)=J(u − 2ν(v)) + JPN . Equation (1.2) is the Jˆ-holomorphic map equation for the map (f,φ); it holds if and only if ¯ (1.3) F =(f,φ) satisfies ∂JˆF =0.

Furthermore, when ν is small Jˆ is tamed by the symplectic formω ˆ = ω ⊕ ωPN ; ≥ 3 | |2 ≤ speciﬁcally, (1.1) implies thatω ˆ((u, v),J(u, v)) 4 (u, v) whenever ν ∞ 1/4. With this tamed condition there are standard elliptic estimates on F = (f,φ); see [PW] and [RT1]. In particular, the energy 1 (1.4) E(f,φ)= |df |2 + |dφ|2 2 is related to the topological quantity 1 2 (1.5) E(F )= |dF | = ωˆ = ω([f(B)]) + ωPN ([φ(B)]) 2 F (B) 3 ≤ ≤ 4 by 4 E(F ) E(f,φ) 3 E(F ). Consequently, we will assume throughout that ν ∞ ≤ 1/4. A stable map isa(J, ν)-holomorphic map for which the energy E(f,φ)is positive on each component Bi; this means that each Bi is either a stable curve or the restriction of f to Bi is nontrivial in homology. A stable map F =(f,φ) is irreducible if F −1(F (x)) = {x} for generic points x (this is automatically true if all the domain components are stable). More generally, a stable map F =(f,φ)isadmissible if it is irreducible when restricted to the union of its unstable domain components Bi with 0 Euler characteristic of the domain. For all (J, ν) the moduli space Mg,n(X, A) of all stable maps is compact, and stabilization and evaluation at the marked points deﬁne a continuous map

st×ev n Mg,n(X, A) −→ Mg,n × X where we make the conventions that Xn is a single point when n = 0 and that in the unstable range (i.e. when 2g − 3+n<0), we take

M0,n = M0,3 for n ≤ 2 and M1,0 = M1,1.

Note that, with this convention, the complex dimension of Mg,n in the stable range is 3g − 3+n while in the unstable range it is g. When all maps in the moduli space are admissible for generic (J, ν) the image of Mg,n(X, A) specifies a homology class called the Gromov-Witten invariant n GWX,A,g,n ∈ H∗(Mg,n × X ). M These invariants can be assembled into a single invariant by setting = M g,n g,n, and introducing variables λ to keep track of the Euler class and tA satisfying tAtB = tA+B to keep track of A. The total GW invariant of (X, ω) is then the formal series 1 − (1.7) GW = GW t λ2g 2 X n! X,A,g,n A A,g,n whose coefficients are multilinear functions on H∗(M) ⊗ T∗(X) where T∗(X) denotes the total tensor algebra T(H∗(X)). This in turn defines the “Gromov- Taubes” invariant GWX GTX = e whose coefficients count holomorphic curves whose domains need not be connected (as occur in [T]). In summary, the results of [RT1] and [RT2] show that generically the moduli spaces of irreducible stable maps are orbifolds and the GW invariants are defined when all maps in Mg,n(X, A) are admissible for generic (J, ν). In practice it is convenient to assume that (X, ω)issemipositive: there are no spherical classes A ∈ H2(X) with ω(A) > 0 and 0 < 2KX A ≤ dim X − 6. For semipositive manifolds all moduli spaces are generically admissible, so that the GW and GT invariants are defined for all n, g and A. On the other hand, one can sometimes show that Mg,n(X, A) is admissible for specific n, g and A even though the manifold is not semipositive.

Remark 1.1 (Local stabilization). If f is a stable map, its domain might have unstable components Bi. However, we can stabilize the domains of stable maps close to f as follows. Introduce k additional marked points on the domain such that each Bi has at least three special points, and so it is stable. Then, THE SYMPLECTIC SUM FORMULA 945 for each of the new points on each Bi, choose cycles Γij whose homological intersection dij with f∗([Bi]) is nonzero, and which are transverse to f and intersect f(Bi) at distinct points. Each stable map close to f then lifts to d = i,j dij maps (counted with sign) that take the new marked points onto ∗ the corresponding cycles Γij. The lift of a (J, ν)-holomorphic map is a (J, π ν)- holomorphic map with a stable domain, where π is the map Mg,n+k → Mg,n that forgets the last k marked points. In the analytic arguments of Sections 5–9, which are local on the space of maps, we will always assume that the domains of the maps have been locally stabilized by this lifting procedure. This allows us to work locally as if all domains were stable, with one important caveat: because the lifted maps are only (J, π∗ν)-holomorphic, the moduli spaces are generically orbifolds only near lifts of irreducible maps f. For that reason, irreducibility appears as a technical assumption in some results in Sections 5–10. It is likely that this irreducibility assumption could be removed by turning on a further generic perturbation ν on U g,n+k. There are several approaches to doing that; see for example in [LT].

The dimension (1.6) is the index of the linearization of the (J, ν)-holomorphic equation, which is obtained as follows. A variation of a map f is specified by a ξ ∈ Γ(f ∗TX), thought of as a vector field along the image, and a variation in the curve C =(B,j,p1,...,pn) is specified by ∈ M ∼ 0,1 ⊗O − (1.8) h TC g,n = Hj TB pi (tensoring with O(−p) accounts for the variation in the marked point p; the correspondence between h and the variation of the map φ is described in Section 4). Let Λ01(f ∗TX) be the vector bundle of all anti-(J, j) linear homomorphisms from TC to f ∗TX. Calculating the variation in the path (1.9) (ft,Ct)= expf (tξ), (j, p1,...,pn)+th one finds that the linearization at (f,C) is the operator ∗ 01 ∗ (1.10) Df,C :Γ(f TX) ⊕ TC Mg,n → Γ(Λ (f TX)) 1 given by Df,C(ξ,h)=Lf,C(ξ)+ 2 Jf∗h with 1 (1.11) L (ξ)(w)= ∇ ξ + J∇ ξ +(J∇ J)(f∗w − Φ (w) − 2ν(w)) f,C 2 w jw ξ f −(∇ξν)(w) where w is a vector tangent to the domain, ∇ is the pullback connection on ∗ f TX, ν(w) means ν(φ∗w), and Φf = ∂J f − ν. Writing L as the sum of its 1 − J-linear component 2 (L JLJ) and its J-antilinear component, we have J (1.12) Lf,C(ξ)(w)=∂f,Cξ(w)+Tf,C(ξ,w). 946 ELENY-NICOLETA IONEL AND THOMAS H. PARKER

J Here ∂f,C is a Jj-linear ﬁrst order operator; so for complex valued functions φ

(1.13) Lf,C(φξ)=∂φ · ξ + φLf,C(ξ)+(φ − φ)Tf,C(ξ,w).

The term Tf,C(ξ,w) is given in terms of the Nijenhuis tensor NJ of J on X and ∇ − ∇ − ∇ ∇ the tensors Tν(ξ,w)=J( ν(w)J)ξ ( ξν)(w) (J Jξν)(w) and J(ξ,Φ) = J(∇ΦJ)ξ +(∇JΦJ)ξ by 1 1 1 (1.14) N (ξ,f∗w − 2ν(w) − Φ (w)) + T (ξ,w)+ ∇J(ξ,Φ (w)). 8 J f 2 ν 4 f

The invariant GWX was generalized in [IP4] to an invariant of (X, ω) relative to a codimension 2 symplectic submanifold V . To define it, we fix a pair (J, ν) which is V -compatible, meaning that along V the normal components of ν and the tensors NJ and Tν satisfy (1.15) (a) TV is J-invariant and νN = 0, and N N ∈ ∈ ∈ (b) NJ (ξ,ζ) = 0 and Tν (ξ,w) = 0 for all ξ TX, ζ TV and w TC. (This is the same as Definition 3.2 in [IP4] after one observes that (b) is automatic for ξ ∈ TV since NJ and Tν(ξ,w)=J[ν, Jξ]+[ν, ξ] are brackets of vector fields in V and hence lie in V .) A stable map into X is called V -regular if f −1(V ) consists of finitely many points x1,...,x none of which are equal to the marked points pi or the nodes. After the xj are numbered, the orders of contact of f with V at the xj define a multiplicity vector s =(s1,...,s) and three associated integers: (1.16) (s)=, deg s = si, |s| = si.

By convention, when = 0 (which corresponds to f −1(V )=∅) we take deg s = 0 and |s| =1. The space of all V -regular maps is the union of components MV ⊂ M (1.17) χ,n,s(X, A) χ,n+(s)(X, A) labeled by vectors s of length = (s) (here χ is the Euler characteristic of the domain). This has a compactification that comes with evaluation maps MV → M × n ×HV (1.18) εV : χ,n,s(X, A) χ,n+(s) X X,A,s. Here Mχ,n is the space of curves with finitely many components, Euler class χ HV and n marked points, and X,A,s is the ‘intersection-homology’ space described HV → × in Section 5 of [IP4]. There is a covering map ε : X,A,s H2(X) Vs whose first component records the class A and whose component in the space ∼ (s) Vs = V records the image of the last (s) marked points. This covering is a necessary complication to the definition of relative GW invariants. THE SYMPLECTIC SUM FORMULA 947

The complication occurs because of “rim tori”. A rim torus is an element of

(1.19) R = ker (ι∗ : H2(X \ V ) → H2(X)) where ι is the inclusion. Each such element can be represented as π−1(γ) where π is the projection SV → V from the boundary of a tubular neighborhood of V (the “rim of V ”) and γ : S1 → V isaloopinV . The group R is the group of deck transformations of the covering R−→HV HV X = X,A,s A,s  (1.20) ε H2(X) × Vs. s HV When there are no rim tori (as is the case if V is simply connected), X,A,s reduces to Vs and the evaluation map (1.18) is more easily described. MV The tangent space to χ,n,s(X, A) is modeled on ker Ds where Ds is N the restriction of (1.10) to the subspace where ξ has a zero of order si at the marked points xi, i =1,...,. For generic (J, ν) as above, cokerDs =0 MV ∗ at irreducible maps and therefore the irreducible part χ,n,s(X, A) of the moduli space is an orbifold with (1.21) χ dim MV (X, A)=−2K A + (dim X − 6)+2n − 2(deg s − (s)). χ,n,s X 2 With this understood, the deﬁnition of the relative GW invariant paral- lels the above deﬁnition of GWX . A stable V -regular map is called (X, V )- admissible if it is irreducible when restricted to the union of its unstable domain components Bi with

(1.22) 0 0, and 0 < · ≤ 1 − { · } KX A + A V 2 (dim X 6) + min A V,2 . Any map from a connected unstable domain Bi satisfying (1.22) is the composition of a covering map with an irreducible map f0 with unstable rational domain which represents a class A satisfying 0

When the moduli spaces are generically admissible the image moduli space under (1.18) carries a homology class which, after summing on χ, n and s, can be thought of as a map V T∗ −→ M×HV Q (1.23) GWX,A : (X) H∗( X ; [λ]). This gives the expansion 1 − (1.24) GWV = GWV t λ2g 2 X (s)! X,A,g,s A A, g ordered seqs s · deg s=A V T∗ → M×HV whose coeﬃcients are (multi)-linear maps (X) H∗ X,A,s (divid- ing by (s)! eliminates the redundancy associated with renumbering the last marked points). The corresponding relative Gromov-Taubes invariant is again given by

V V GWX (1.25) GTX = e . V After imposing constraints one can expand GTX in power series. That is done in the appendix under the assumption that there are no rim tori.

2. Symplectic sums

Assume X and Y are 2n-dimensional symplectic manifolds each containing symplectomorphic copies of a codimension two symplectic submanifold (V,ωV ). Then the normal bundles are oriented, and we assume they have opposite Euler classes:

(2.1) e(NX V )+e(NY V )=0. ∗ We then fix, once and for all, a symplectic bundle isomorphism ψ :(NX V ) → NY V . These data determine a family of symplectic sums Zλ = X#V,λY para- metrized by λ near 0 in C; these have been described in [Gf] and [MW]. In fact, this family fits together to form a smooth 2n + 2-dimensional symplectic manifold Z that fibers over a disk. In this section we will construct Z and describe its properties.

Theorem 2.1. Given the above data, there exists a 2n +2-dimensional symplectic manifold (Z, ω) and a fibration λ : Z → D over a disk D ⊂ C. The center fiber Z0 is the singular symplectic manifold X ∪V Y , while for λ =0, the fibers Zλ are smooth compact symplectic submanifolds — the symplectic connect sums.

This displays the Zλ as deformations, in the symplectic category, of the singular space X ∪V Y as in Figure 1. For λ = 0 these are symplectically isotopic to one another and to the sums deﬁned in [Gf] and [MW]. THE SYMPLECTIC SUM FORMULA 949

The proof of Theorem 2.1 involves the following construction. Given a complex line bundle π : L → V over V , fix a hermitian metric on L, set ρ(x)= 1 | |2 ∈ 2 x for x L, and choose a compatible connection on L. The connection defines a real-valued 1-form α on L\{zero section} with α(∂/∂θ) = 1 (identify the principal bundle with the unit circle bundle and pull back the connection form by the radial projection) and the curvature of α defines a 2-form F on V with π∗F = dα. Then the form ∗ ∗ (2.2) ω = π ωV + ρπ F + dρ ∧ α extends across the zero section and is S1-invariant, closed, and nondegenerate for small ρ. The moment map for this S1 action x → eiθx is the function −ρ because i ∂ ω = i ∂ (dρ ∧ α)=−dρ. ∂θ ∂θ ∗ ∗ 1 | |2 The dual bundle L has a dual metric, a radial function ρ (y)= 2 y , and connection α∗ with dα∗ = −π∗F . This gives a symplectic form similar to (2.2) on L∗ and hence one on π : L ⊕ L∗ → V , namely ∗ ∗ ∗ ∗ ∗ (2.3) ω = π ωV +(ρ − ρ ) π F + dρ ∧ α + dρ ∧ α . Below, we will denote points in L ⊕ L∗ by triples (v, x, y) where v ∈ V and (v, x, y) is a point in the fiber of L ⊕ L∗ at v. This space has (2.4) (a) a circle action (v, x, y) → (v, eiθx, e−iθy) with Hamiltonian t(v, x, y)=ρ∗ − ρ, (b) a natural S1-invariant map L ⊕ L∗ → C by λ(z, x, y)=xy ∈ C.

Proof of Theorem 2.1. Fix (ω, J, g)onL = NX V as above. Using ψ and the Symplectic Neighborhood Theorem, we symplectically identify a neighborhood of V in X with the disk bundle of radius ε ≤ 1inL and a neighborhood of V in Y with the ε-disk bundle in L∗. We assume that ε =1; the general case then follows by rescaling (ω, J, g). Let D denote the disk of radius δ<1/2inR2 with the symplectic form ωD = rdrdθ. The space Z is constructed from three open pieces: two ends EndX =(X \ V ) × D, and EndY =(Y \ V ) × D and a “neck” modeled on the open set ∗ (2.5) U = { (v, x, y) ∈ L ⊕ L ||x|≤1, |y|≤1}. These are glued together by the diﬀeomorphisms ∗ (2.6) ψX : U \ L → EndX by (v, x, y) → ((v, x),λ(v, x, y)),

ψY : U \ L → EndY by (v, x, y) → ((v, y),λ(v, x, y)). This deﬁnes Z as a smooth manifold. The function λ extends over the ends as the coordinate on the D factor, giving a projection λ : Z → D whose ﬁbers are smooth submanifolds Zλ for small λ =0. 950 ELENY-NICOLETA IONEL AND THOMAS H. PARKER

∗ Y × D L 6 6 1 t Z − ¨ λ (1 δ) ¨¨ ¡ ¡ ¡ 6 & X × D δ r - ? V (1 − δ) 1 L

Figure 3. Construction of Zλ

In the overlap region of U where (1 − δ) ≤|x|≤1 and |y|≤δ the form ∗ ∗ (2.3) is ωX + d(ρ α ) (this ωX is the pull-back of the symplectic form on X to ∗ ∗ ⊕ ∗ L ), and ψX (ωX ωD)=ωX + λ ωD because of the symplectic neighborhood identification. But 2λ∗(rdrdθ)=d(|λ|2 λ∗dθ)=4d(ρρ∗ λ∗dθ) and, since dθ is the connection on the trivial bundle L ⊗ L∗, λ∗dθ is the connection form on L ⊕ L∗, namely α + α∗.Thus ∗ ∗ ∗ λ ωD = d(ρ η) where η =2ρ (α + α ). ∗ ∗ ∗ We can then smoothly merge λ ωD into d(ρ α ) by replacing η byη ˆ = βη + (1 − β)α∗ where β = β(|x|) is an appropriate cutoff function with |dβ| < 2/δ. In this overlap region 2ρ lies between (1 − δ)2 and 1, dα and dα∗ are bounded, and |α|≤2 and |α∗|≤2/δ. It follows that α∗ − η satisfies |α∗ − η|≤C and |d(α∗ − η)|≤C/δ, and hence ∗ ∗ ∗ ∗ |d(ρ ηˆ) − d(ρ η)| = |d [ρ (1 − β)(α − η)] |≤Cδ.

∗ Because ωX + d(ρ η) is nondegenerate and δ is as small as desired, the above ∗ inequality shows that on this overlap region ωX + d(ρ ηˆ) is closed and nondegenerate for small δ. Thus we have a specific formula extending (2.3) over EndX as a symplectic form whose restriction to the part of Zλ ⊂ EndX with |x|≥1 is the original symplectic form ωX on X ×{λ}. Repeating the construction on the Y side yields a global symplectic form ω on Z whose restriction on ∗ the Y side is ωY . Finally, along Zλ ∩ U we have α = −α, so that ω restricts to ∗ ∗ ωλ = π ωV − tπ F − dt ∧ α with t as in (2.4). Thus after possibly making δ smaller, we have a fibration λ : Z → D with symplectic fibers. THE SYMPLECTIC SUM FORMULA 951

This construction shows that the neck region U of Z has a symplectic S1 1 | |2 −| |2 action with Hamiltonian t = 2 ( y x ). This action preserves λ, and so restricts to a Hamiltonian action on each Zλ. In fact, t gives a parameter along the neck, splitting each Zλ into manifolds with boundary − ∪ + Zλ = Zλ Zλ − ∩ ∩ ≤ where Zλ is Zλ EndX together with the part of Zλ U with t 0. From this decomposition we can recover the symplectic manifolds X and Y up to isotopy in two ways:

→ − + \ (1) as λ 0, the interior of Zλ (respectively Zλ ) converges to X V (respectively Y \ V ) as symplectic submanifolds of Z,or

− + (2) X (resp. Y ) is the symplectic cut of Zλ (resp. Zλ )att =0.

Thus we have collapsing maps

X YZλ

(2.7) π0 πλ Z0 and πλ is a deformation equivalence on the set where t =0. The next step is to define a Riemannian metric and an almost complex structure on (Z, ω) so that the triple (ω, J, g) is compatible in the sense of (1.1). We begin by specifying such a triple on the total space of L → V . For each small ρ>0 we can extend the symplectic form ωρ = ωV + ρF to a compatible triple (ωρ,Jρ,gρ)onV with gρ = gV + O(ρ). At each p =(v, x) ∈ L with 1 | |2 ⊕ ρ = 2 x = 0, there is a splitting TpL = H Lp into a horizontal subspace H =kerdρ ∩ ker α and a vertical subspace ker π∗ identified with Lp, and by (2.2) this splitting is ω-orthogonal. Using this splitting, define a metric ∗ on TpL by g0 = π gρ ⊕ gL. Starting from this g0 and ω, the polarization procedure described in the appendix of [IP4] produces a pair (J, g) compatible with ω which respects the splitting and which extends across the zero section. In fact, around each point of V there is a local trivialization of L in which 2 g = gV ⊕ gC + O(|x| ). Applying the same construction on L∗ with the dual connection and taking the direct sum, we have a compatible structure (ω, J, g) on a neighborhood of the zero section V ⊂ L ⊕ L∗. That structure locally agrees with the product structure on V × C × C to second order along V ,soisV -compatible as in (1.15), the second fundamental form h of V is zero, and the Nijenhuis tensor satisfies NJ (ξ,v),η = 0 for all v ∈ TV and ξ,η normal to V . In particular, the pair (J, ν) with ν = 0 satisfies condition (i) of the following definition. 952 ELENY-NICOLETA IONEL AND THOMAS H. PARKER

Deﬁnition 2.2. Let J (Z) be the set of extensions of the symplectic structure ω on Z to a compatible data (ω, J, g, ν) with ν ∞ < 1/4 as in (1.4) and so that (i) along V we have (1.15ab) and the second fundamental form of V ⊂ Z vanishes, and (ii) Zλ is J-invariant for all λ.

Now consider compatible structures (ωX ,JX ,gX ,νX )onX and (ωY ,JY , gY ,νY )onY which satisfy condition (i) above. If these agree along V , they deﬁne a compatible structure on Z0 and the following lemma constructs corresponding elements of J (Z).

Lemma 2.3. Given ε>0 and structures on X and Y as above, there exists an element (ω, J, g, ν) ∈J(Z) which agrees with the given data on Z0 \ U(ε) and along V . In particular, J (Z) is nonempty.

Proof. We have just seen that for small ε there are data (ω, J, g,0) on U(ε) satisfying (i) of Definition 2.2. These data also satisfiy (ii) at points p =(v, x, y) ∈ U(ε), as follows. Each path vt in V starting at v has a horizontal ∗ lift (vt,xt,yt)inL ⊕ L with initial point p. Along the lift, λ = x · y and λ = ∇v˙ x · y + x ·∇v˙ y =0,soλ is constant. Thus each horizontal lift lies in Zλ for λ = λ(p). It follows that TpZλ is the sum of the horizontal subspace at p ∗ 2 and the tangent space to the complex curve xy = λ in the fiber (L⊕L )v = C . Both of those subspaces are J-invariant, so (ii) holds in U(ε). We can extend g on U(ε/2) to a Riemannian metric g0 on Z which agrees with the given product metrics Z0 \ U(ε). By Lemma A.1 of [IP4], (ii) is equivalent to the condition that the symplectic normal Np to TpZλ is equal to the metric normal. This already holds on U(ε) and on Z0 \ V , and we can choose g0 so that it holds everywhere. As above, applying the polarization procedure to g0 and ω yields a pair (J, g) compatible with ω which still respects the decomposition TZλ ⊕ N and which has g = g0 in the regions where g was already compatible with ω. Finally, we can extend ν on Z0 \ B(ε) arbitrarily.

We will work with structures in J (Z) throughout the analytical sections of this paper. We conclude this section with a useful lemma comparing the canonical class of the symplectic sum with the canonical classes KX and KY of X and Y .

Lemma 2.4. If A ∈ H2(Zλ; Z), λ =0, is homologous in Z to the union C1 ∪ C2 ⊂ X ∪V Y of cycles C1 in X and C2 in Y , then

KZλ [A]=KZ [A]=KX [C1]+KY [C2]+2β · · where β is the intersection number V [C1]=V [C2]. In particular, KZλ [R]=0 for any rim torus R (cf. (1.19)). THE SYMPLECTIC SUM FORMULA 953

Proof.Forλ = 0, the normal bundle to Zλ has a nowhere-vanishing section ∂/∂λ. Thus the canonical bundle of Zλ is the restriction of the canonical bundle of Z, giving

KZλ [A]=KZ [A]=KZ [C1]+KZ [C2].

Along X ⊂ Z0 the tangent bundle to Z decomposes as ∗ ∗ ∼ ∗ −1 TZ = TX ⊕ π ψ NY V = TX ⊕ π (NX V ) where π is the projection NX V → V . But the Poincar´e dual of V in X, 2 ∗ regarded as an element of H (X), is the Chern class c1(π NX V ). Since the canonical class is minus the ﬁrst Chern class of the tangent bundle we conclude that KZ [C1]=KX [C1]+V · [C1] and similarly on the Y side.

3. Degenerations of symplectic sums

The Gromov-Witten invariants of the symplectic sum Zλ are defined in terms of stable pseudo-holomorphic maps from complex curves into the Zλ. The basic idea of our symplectic sum formula is to approximate the maps in Zλ by certain maps into the singular space Z0. The first step is a limiting argument. The key point is that, by the construction in Section 2, the spaces {Zλ} are embedded in a compact almost Kähler manifold Z — the closure of a neighborhood of the central fiber of the family Z. Hence the “Gromov Compactness Theorem” implies that sequences of (J, ν)-holomorphic maps into Zλ limit to maps into Z0 (after passing to subsequences). This section gives a description of the maps into Z0 which arise as limits of δ-flat stable maps into the Zλ as λ → 0. The ‘δ-flat’ condition, defined below, ensures that the limit has no components mapped into V . Fix a small δ>0. Given a map f into Zλ, we can restrict attention to that part of the image that lies in the ‘δ-neck’ 2 2 (3.1) Zλ(δ)={z =(v, x, y) ∈ Zλ |||x| −|y| |≤δ}. This is a narrow region symmetric about the middle of the neck in Figure 3. The energy of f (more precisely of (f,φ)) in this region is 1 (3.2) Eδ(f)= |df |2 + |dφ|2 2 −1 where the integral is over f (Zλ(δ)). By Lemma 1.5 of [IP4] there is a constant αV < 1, depending only on (JV ,νV ) such that every component of every stable (JV ,νV )-holomorphic map into V has energy

(3.3) E(f) ≥ αV . 954 ELENY-NICOLETA IONEL AND THOMAS H. PARKER

Definition 3.1 (δ-flat map). A stable (J, ν)-holomorphic map f into Z is δ-flat if the energy in the δ-neck is at most half αV , that is δ (3.4) E (f) ≤ αV /2.

Note that a stable δ-flat map into Z0 cannot have any components (not even ghosts) lying in V and is thus V -regular in the sense of [IP4]. For each small λ, let MV,δ χ,n(Zλ,A) denote the set of δ-flat maps in Mχ,n(Zλ,A). These are a family of subsets of the space of stable maps into Z and we write MV,δ (3.5) lim χ,n(Zλ,A) λ→0 for the set of limits of sequences of δ-flat maps into Zλ as λ → 0. Because (3.4) is a closed condition this limit set is a closed subspace of Mχ,n(Z0,A). The remainder of this section is devoted to a precise description of the space (3.5).

Lemma 3.2. Each element of (3.5) is a stable map f to Z0 = X ∪V Y with no components of the domain mapped entirely into V .

Proof. By the compactness theorem for (J, ν)-holomorphic maps (cf . §1 of [IP4]) each sequence in (3.5) has a subsequence fk converging in the space of stable maps Mχ,n(Z, A) to a limit f : C → Z. In particular, the images converge pointwise, and thus lie in Z0. Suppose that the image of some component Ci of C lies in V . Then the δ restriction fi of f to that component satisﬁes E(fi) ≤ E (f). Furthermore, by Theorem 1.6 of [IP4] the sequence fk (after precomposing with diﬀeomor- 0 δ δ phisms) converges in C and in energy, so E (f) = lim E (fk) ≤ αV /2. This contradicts (3.3).

We can be very specific about how the images of the maps in (3.5) hit V . By Lemma 3.2 and Lemma 3.4 of [IP4], at each point p ∈ f −1(V ) the normal d component of f has a local expansion a0z + ... . This defines a local ‘degree of contact’ with V (3.6) d = deg(f,p) ≥ 1 and implies that f −1(V ) is a finite set of points. By restricting f to one −1 component Ci of C and removing the points f (V ), one obtains a map from a connected domain to the disjoint union of X \ V and Y \ V . Thus the X components of C are of two types: those components Ci whose image lies in Y X, and those components Ci whose image lies in Y . We can therefore split f into two parts: the union of the components whose image lies in X defines a map f1 : C1 → X, from a (possibly disconnected) curve C1, and the remaining components define a similar map f2 : C2 → Y . THE SYMPLECTIC SUM FORMULA 955

Lemma 3.3. f −1(V ) consists of nodes of C. For each node x = y ∈ f −1(V ) deg(f1,x) = deg(f2,y).

Proof. The local degree (3.6) is a linking number. Speciﬁcally, let NX V be a tubular neighborhood of V in X and let µX be the element of H1(NX V \ V ) represented by the oriented boundary of a holomorphic disk normal to V .If µY is the corresponding element on the Y side, then µX = −µY in H1 of the −1 neck Zλ(δ). For each point x in f1 (V ) and each small circle Sε around x, the local degree d satisﬁes d · µ =[f1(Sε)]. 1 If x isnotanodeofC then by Theorem 1.6 of [IP4] fk converges to f1 in C in a disk D around x. But then for large k, d·µ =[f(Sε)] = [fk(Sε)] = [fk(∂D)] = 0, contradicting (3.6). Next consider a node x = y of C which is mapped into V . Choose holomorphic disks D1 = D(x, ε) and D2 = D(y, ε) that contain no other −1 points of f (V ) and let Si = ∂Di. Then S1 ∪ S2 bounds in C, so that 0 [fk(S1)] + [fk(S2)]=0inH1 of the neck Zλ(δ). Again, fk → f in C , which implies that 0 = [f(S1)] + [f(S2)] = d1µ1 + d2µ2 where µi is either µX or µY , depending on which side f(Si) lies. Since di > 0 the only possibility is that x = y is a node between a component in X and one in Y and d1 = d2.

Lemmas 3.2 and 3.3 show that each map f in the limiting set (3.5) splits into (J, ν)-holomorphic maps f1 : C1 → X and f2 : C2 → Y . Ordering −1 the nodes in f (V ) gives extra marked points x1,...,x on C1 and matched y1,...,y on C2 with si = deg xi = deg yi. Furthermore, the Euler characteristics χ1 of C1 and χ2 of C2 satisfy

(3.7) χ1 + χ2 − 2 = χ.

. . X x1 y1 X . Y . . x y . 2 2 f . . . x3 y3

Figure 4. The map f0 =(f1,f2)intoZ0 = X ∪V Y 956 ELENY-NICOLETA IONEL AND THOMAS H. PARKER

We can now give a global description of how the limit maps f in (3.5) are assembled from their components f1 and f2. First, consider how the domain curves ﬁt together in accordance with (3.7). Given bubble domains C1 and C2, not necessarily connected or stable, with Euler characteristics χi and ni + marked points, we can construct a new curve by identifying the last marked points of C1 with the last marked points of C2, and then forgetting the marking of these new nodes. After possibly adding more marked points to stabilize any unstable components, this process is the same as the standard attaching map M × M −→ M (3.8) ξ : χ1,n1+ χ2,n2+ χ1+χ2−2,n1+n2 whose image is a subvariety of complex codimension . Taking the union over all χ1,χ2,n1 and n2 gives a (stabilized) attaching map ξ : M×M→M for each . Second, consider how the maps ﬁt together along V . The evaluation map × × MV ×MV εV−→εV H V ×HV ε−→2 ε2 × evss : χ1,n1,s(X) χ1,n2,s(Y ) X Y Vs Vs. records the intersection points with V and the pair (f1,f2) lies in the space

V V def −1 (3.9) M (X) × M (Y ) =evss (∆s). evs

where ∆s is the diagonal ∆s ⊂ Vs × Vs. HV × HV × −1 HV HV Denote by X ε Y =(ε2 ε2) (∆) the ﬁber sum of X and Y along the evaluation map ε2, where ∆ = ∆s. Then we have a well deﬁned map s HV × HV → (3.10) g : X Y H2(Z) ε which describes how the homology-intersection data of f1 and f2 determine the homology class of f.

Lemma 3.4. For generic (J, ν) the irreducible part of the space (3.9) is a V,δ smooth orbifold of the same dimension as Mχ,n(Zλ,A), given by (1.6). MV ∗ MV ∗ Proof. The dimensions of χ1,s(X, A1) and χ2,s(Y,A2) are given by (1.21). A small modiﬁcation of the proof of Lemma 8.5 of [IP4] shows that the evaluation map at the last = (s) marked points (i.e. the intersection points with V ) is transversal to the diagonal ∆ ⊂ V ×V , imposing dim V = (dim X −2) conditions. Thus the irreducible part of (3.9) is a smooth orbifold of dimension 1 −2K [A ] − 2K [A ] − 4 deg s − (dim X − 6)(χ + χ − 2)+2n. X 1 Y 2 2 1 2 The lemma follows by comparing this with (1.6) using (3.7), Lemma 2.4, and the fact that deg s = A1 · V = A2 · V . THE SYMPLECTIC SUM FORMULA 957

Finally, note that renumbering the pairs (xi,yi) of marked points defines an action of the symmetric group S on (3.9) and the limit maps in (3.5) correspond to elements in the quotient. Moreover, after ordering the double points along V the limit set (3.5) is a subset of the set V V (3.11) Kδ ⊂ M (X) × M (Y ) ev s s consisting of δ-flat maps into Z0 with fixed data (χ, n, A).

Remark 3.5. Observe that the set Kδ is compact and that the multiplicity vector s =(s1,...s) is constant on components of Kδ. Indeed, as in the proof of Lemma 3.2, any sequence {fn}∈Kδ has a subsequence converging to a V -regular map f0 into Z0. Because the energy (3.2) is continuous in the stable map topology the limit is δ-ﬂat, and thus lies in Kδ. Furthermore, after lifting to the normalization of the domain, stable map convergence implies that the −1 ∈ −1 marked points xk,n of fn (V ) converge to points xk f0 (V ) distinct from the ∞ nodes. This in turn means that fn converges to f0 in C in a neighborhood of each xk. Consequently the order of contact sk of f0 with V at xk is at least sk. Since the total intersection sk = A · V is preserved in the limit and all multiplicities are positive, we conclude that s = s. In particular, no new intersection points with V arise in the limit.

V,δ 0 Since the δ-flat maps in M (Zλ) are C close to δ-flat maps into Z0 for small λ there is a decomposition MV,δ MV,δ (Zλ)= s (Zλ) S(s) s as a union of components labeled by ordered sequences s =(s1,s2 ...). As in the proof of Lemma 3.3, these si are local winding numbers of the (s) vanishing cycles Sε. In that form the labeling extends to all continuous maps 0 C close to δ-flat maps into Z0. Thus for small λ MV,δ ⊂ s (Zλ) Maps(Zλ) where Maps(Zλ), the “space of labeled maps”, is the set of labeled continuous 0 maps into Zλ which are C close to δ-flat maps into Z0. Thus with this notation, the statements of Lemmas 3.2 and 3.3 translate into the commutative diagram V V V,δ M (X) × M (Y ) ←− lim Ms (Zλ) evs λ  s  s  (3.12) ξ×g (M×M) × HX × HY −→ M×H2(Z). ε 958 ELENY-NICOLETA IONEL AND THOMAS H. PARKER

The top arrow shows how the maps that arise as limits of δ-flat maps decompose into pairs (f1,f2)ofV -regular maps into X and Y , while the bottom arrow keeps track of the domains and homology classes (the vertical maps arise from (1.18) in the obvious way). One then expects the top arrow in (3.12) to be a diffeomorphism for each s and each side to be a model for the stable maps into Zλ for that s. The analysis of the next six sections will show that this is true after passing to a finite cover. The necessity of passing to covers is dictated by the clustering phenomenon mentioned in the introduction: when s>1 each curve in Z0 is close (in the stable map topology) to a cluster of curves in Zλ for small λ, and these coalesce as λ → 0. To distinguish the curves within a cluster and, indeed even to verify this statement about clustering, it is necessary to use stronger norms and distances — strong enough that the distances between the maps within a cluster are bounded away from zero as λ → 0. The maps in a cluster can then be thought of as the fiber of a covering of the space of limit maps. The next three sections introduce the required norms and construct a first version of the covering. The first step is to define an appropriate distance function on the space of stable curves.

4. The space of curves

One can measure the distance between stable curves using a metric on the moduli space Mg,n of curves. However, it is often more convenient to fix a diffeomorphism of the curves, regard the two curves as two complex structures on a single 2-manifold, and measure the distance between the complex structures using a Sobolev norm. In this section we take that approach to define a metric and distance function on Mg,n. Our metric is designed so that a neighborhood of the image of the attaching map (3.8) is obtained by gluing cylindrical ends of the spaces Mg,n. It is a complete metric on Mg,n \N where N is the set of all nodal curves; in particular it is stronger than the Weil-Petersson metric. To simplify the exposition we will assume in Sections 4–9 of this paper that stable domains either have no nontrivial automorphisms or come with Prym structures as defined in [Loo]. Prym structures define finite covers of the Deligne-Mumford spaces Mg,n which are smooth projective varieties. In particular, the corresponding universal curves are smooth and projective, so can be used to define ν as in (1.1). The construction starts by fixing a Riemannian metric gU on the universal π curve U g,n → Mg,n compatible with the complex structure. In U g,n the ‘special points’ (marked points and nodes) are distinct and hence, by compactness, are separated by a minimum distance. After rescaling the metric we can assume that the separation distance is at least 4. We also fix a smooth positive function THE SYMPLECTIC SUM FORMULA 959

ρˆ on U g,n equal to the distance to the node in a neighborhood of each node. Finally, we replace gU by a conformal metric that is singular along the nodal points locus, namely −2 (4.1) g =ˆρ gU .

To understand the geometry of this metric we focus attention on a small ball U in the set N of nodal curves and construct a local model for a neigh- −1 borhood of π (U)inU g,n. The curves C0 in U have normalizations C0 with n marked points plus pairs of marked points xk,yk with C0 obtained by identifying each xk with yk. For each k we ﬁx local coordinates {zk} around xk and {wk} around yk on the C0 ∈ U in which the metric (4.1) is Euclidean. We can use the construction of Section 2 to form a family of symplectic sums of curves; for details see [Ma]. The result is a holomorphic ﬁbration F with maps F−−−→U  g,n   (4.2)

U × D −−−→ Mg,n where D is the unit disk in C . The ﬁber of F over (C0,µ) is a curve C0(µ) given by zkwk = µk in disjoint balls Bk centered on the nodes. Outside the union of the Bk we can ﬁx a trivialization of F which respects the marked points. The horizontal arrows in (4.2) are biholomorphic away from the curves with nontrivial automorphisms and biholomorphic everywhere for curves with Prym structures.

Remark 4.1. The parameters {µk} are intrinsically elements of the bundle L ⊗L ∗ (4.3) ( k k) k=1 L L where k and k are the relative cotangent bundles to C0 at xk and yk respectively. Thus the ﬁbration (4.3) models a tubular neighborhood of U ⊂N in Mg,n.

Fix a metric on F whose restriction to each ﬁber is Euclidean in the coordinates (zk,wk) on each Bk, scaling the metric so that each Bk has radius at least 4. Inside Bk, the induced metric on C0(µ)is | |2 2 2 µ 2 2 2 (4.4) gµ =Re(dz + dw ) = 1+ 4 dr + r dθ zw=µ r 2 2 2 where r = |z| and the distance to the node in Bk is ρ = |z| + |w| = 2 2 2 −2 r + |µ| /r . Switching to the conformal metric g = ρ gµ as in (4.1), and 1 assuming that µk = 0, we can identify C0(µ) ∩ Bk(2) with [−Tk,Tk] × S 960 ELENY-NICOLETA IONEL AND THOMAS H. PARKER t by writing r = |µk| e with Tk = | log(2/ |µk|) |. In these coordinates 2 ρ =2|µk| cosh(2t) and −2 −1 2 2 2 2 (4.5) g = ρ gµ = r dr + dθ = dt + dθ .

Thus with this metric on F the curves C0(µ) have necks which are isometric to cylinders of radius one and length 2Tk with Tk →∞as µk → 0. Because the top map in (4.2) is holomorphic with bounded differential, its restriction to each fiber is conformal and the conformal factor is bounded. Consequently, the PDE results of the next several sections, all of which involve only local considerations in the space U g,n, can be done in the model space F using the metric (4.5) and the results will apply uniformly on U g,n. We will henceforth consistently use this metric (4.5) on the domains of holomorphic curves and will no longer distinguish between (4.1) and (4.5). Note that the flatness condition (3.4) continues to hold (after a uniform change of constants) because the energy density is conformally invariant. We next define a metric on U × D in terms of a Sobolev metric on the fibers of F. In the directions tangent to U this will be the Weil-Petersson metric · WP. To describe the metric in the D directions, we fix an -nodal curve C0 and consider the restriction F0 of F over {C0}×D . The first step is to construct a diffeomorphism between smooth fibers of F0. Recall that inside Bk(2) the fibers of F0 are given by zw = µk, while outside the union of the Bk(1) F0 has a trivialization which identifies (z, µ/z) with (z,µ /z) when 1 ≤|z|≤2 and (µ/w, w) with (µ/w, w) when 1 ≤|w|≤2; see Figure 3 above and [Ma, p. 626]. ∩ For each smooth fiber Cµ = C0(µ) we can parametrize the necks Cµ Bk(2) | | t+iθ ∈ − × 1 by writing z = µk e as above with (t, θ) [ Tk,Tk] S for Tk = | log(2/ |µk|) |. Given a second smooth fiber Cµ we can similarly parametrize ∩ − × 1 k ∩ → ∩ Cµ Bk(2) by [ Tk,Tk] S and define a map ψµµ : Cµ Bk(2) Cµ Bk(2) by k 1 µ µ ψ (t, θ, µ)= t + η(t) − log ,θ− η(t) arg ,µ µµ 2 µ µ where η(t) is a cutoff function equal to 1 for t ≤−1 and 0 for t ≥ 1. This diffeomorphism between the necks, one readily checks, agrees with the given trivialization on Bk(2) \ Bk(1) and so defines a diffeomorphism Cµ → Cµ . The corresponding infinitesimal diffeomorphism defines lifts of vectors v = (v1,...,v) ∈ TµD to F0: v defines a path µ(s)=µ + sv and a vector field d − 1 vk − · vk v˜ = ψµµ(s) = η Re , η Im ,v ds 2 µk µk s=0 k along Cµ. Going the other way, given any path µ(s) with µk(s) nonzero for all k and s, we can lift the vectorsµ ˙ as above and integrate the lifted vector THE SYMPLECTIC SUM FORMULA 961

→ fields to get diffeomorphisms ψs : Cµ(0) Cµ(s). The variation in the complex d ∗ structure is h1 = ψ j|s=0. Define the metric by ds s 2 2 2 2 2 (4.6) h1 = |∇ h1| + |∇h1| + |h1| dvg Cµ using the norms and volume form of the cylindrical metric (4.5). With this norm the Sobolev inequality sup |h1|≤c h1 holds with a constant uniform in µ (the cylindrical metric on Cµ has an upper bound for curvature and a lower bound on the injectivity radius independent of µ).

Lemma 4.2. On the complement of the nodal set N = {µ |some µk is zero} the Riemannian metric (4.6) on {C0}×D is uniformly equivalent to the metric 2 |dµk| (4.7) 2 . |µk| k

Proof. Calculating h1 = Lv˜j = Lv˜(∂θ ⊗ dt − ∂t ⊗ dθ) by computing the Lie derivatives Lv˜∂θ =[˜v, ∂θ]=0,Lv˜dt = dv˜t, Lv˜∂t =[˜v, ∂t], and Lv˜dθ = dv˜θ, one ﬁnds that   A =Re vk ⊗ − − ⊗ µk h1 = η (B∂t + A∂θ) dt η (A∂t B∂θ) dθ where  B =Im vk . µk Because dη has support in [−1, 1] and the integrals of |dη|, |∇dη|, and |∇2dη| are independent of µ we then have | |2 1 2π | |2 2 2 vk |∇2 |2 |∇ |2 | |2 vk h1 = 2 dη + dη + dη dtdθ = c 2 . |µk| − |µk| k 1 0 k

Deﬁnition 4.3. For h =(h0,h1) ∈ T (U × D ) in the chart (4.2) set 2 2 2 (4.8) (h0,h1) = h0 WP + h1 .

After covering Mg,n by such charts (with ≥ 0) and using a partition of unity, we see that (4.8) gives a Riemannian metric on Mg,n \N, well-deﬁned up to uniform equivalence. We ﬁx a metric in that equivalence class. ∈ × By Lemma 4.2 the metric (4.8) on h =(h0, µ˙ ) TC0(µ)(U D ) is uniformly equivalent to 2 2 ∼ 2 µ˙ k (4.9) (h0, µ˙ ) = h0 WP + . µk k

Furthermore, the metric (4.7) is cylindrical in each coordinate: writing µk = t+iθ −2 2 2 2 2 e the terms of (4.7) gives |µk| Re (dµk) = |d log µk| = dt + dθ . The 962 ELENY-NICOLETA IONEL AND THOMAS H. PARKER corresponding distance function is also cylindrical, so the distance squared t+iθ s+iθ | − |2 | − |2 | |2 between µ = e and µ = e is t t + θ θ = log µk/µk .Thus for -nodal curves C and C the distance function of the metric (4.8) is given up to uniform equivalence by 2 ∼ µ (4.10) dist2 C(µ),C (µ ) = dist2(C, C )+ log k . µk k

Thus the metric (4.8) on Mg,n = Mg,n \N is complete; near the stratum N of curves with nodes it is asymptotic to a cylinder W × (0, ∞) where W is a bundle over N whose fiber is the real torus T corresponding to the bundle (4.3). Finally, observe that this geometry leads to a nonstandard compactification of Mg,n : identify the end W × (0, ∞) with W × (0, 1) and compactify to W × (0, 1] . This “cylindrical end compactification” projects down to the Deligne-Mumford compactification Mg,n so that the fiber along the nodal stratum N is a copy of W.

5. Renormalization at the nodes

In this section we will consider a sequence of δ-ﬂat (J, ν)-holomorphic maps → → (5.1) fn : Cn Zλn with λn 0. By the Compactness Theorem for holomorphic maps into Z these converge to a limit map f0 from a nodal curve C0 to Z0 as described in Section 3; the convergence is in L1,2,inC0, and in C∞ on compact sets in the complement of the nodes. We will reﬁne this by constructing renormalized maps fˆn around each node that is mapped into V , and proving convergence results for the renormalized maps. This gives detailed information about how the original maps fn are converging in a neighborhood of the nodes. As in Section 3, the limit domain C0 is the union of (not necessarily connected) curves C1 and C2 which intersect at nodes, and f0 decomposes into maps f1 : C1 → X and f2 : C2 → Y whose images meet along V with contact vector s =(s1,...s). Thus, after regarding C1 and C2 as disjoint curves, there are points xk ∈ C1 and yk ∈ C2, k =1,...,, so that f1 and f2 contact V of order sk at the point qk = f1(xk)=f2(yk)inV . For short, we simply write

fn → f0 =(f1,f2) ∈Kδ ⊂Ms ×ev Ms. where Kδ is the compact set introduced in (3.11). All the estimates in the next several sections will be uniform on Kδ. For the rest of this section we will focus attention on the restrictions of the maps (fn,φn):Cn → Z × U to the neck near a ﬁxed node xk = yk of C0 ⊂ U. THE SYMPLECTIC SUM FORMULA 963

There we have coordinates (z,w) centered on the node described before (4.4). However, it is often more convenient to parametrize the graphs zw = µn by the cylindrical coordinates (t, θ) used in the previous section. Thus we write t+iθ (5.2) φn(t, θ)=(z,w)=(z,µn/z) where z = |µn| e and, as in (1.2), regard (fn,φn) as a map 1 (5.3) (fn,φn):[−Tn,Tn] × S → Zλ × U g,n n with Tn = | log(2/ |µn|) |→∞as n →∞and with the cylindrical metric (4.5) on the domain. We will use the cylindrical metric in all of our PDE estimates. Since (4.5) is a conformal change of metric we still have fn → f0 in C0 and in energy. Our goal is to bootstrap from there. We also choose coordinates on Z around the image point q = qk ∈ V of the 2n−2 n−1 node as follows. Fix a J-equivariant identiﬁcation of TqV with R = C and extend that to normal coordinates {vi} around q in V . Then identifying Lq with C, taking the direct sum with the dual, and parallel translating along radial lines in the v coordinates, we obtain coordinates (x, y):L⊕L∗ → C⊕C. This gives the desired coordinates (5.4) (v, x, y) because, as in Section 2, L ⊕ L∗ is the normal bundle to V in Z. In these coordinates the almost complex structure J on Z agrees with the standard n−1 complex structure J0 on C ⊕ C ⊕ C at the origin, and J has the form i JV ⊕ JC ⊕ JC along V . Notice that the v are real-valued while x and y are complex-valued, and in these coordinates xy = λ. In these coordinates our maps have components fn =(vn,xn,yn) where vn is a map into V , xn = x ◦ fn is the coordinate normal to V in X, yn is the coordinate of fn normal to V in Y , and Zλ is locally the graph of xy = λ. The expansions of f0 provided by Lemma 3.4 of [IP4] and the matching condition of Lemma 3.3 show that

v sk x sk y (5.5) f0(z,w)=(h ,akz (1 + h ),bkw (1 + h )) where |(hv,hx,hy)|≤cρ. In the next lemma we ﬁx F =(f,φ) as in (5.3) and estimate the neck energy 1 T 2π E(F, T)= |dF |2 dt dθ. 2 −T 0

Lemma 5.1. For each (J, ν) ∈J(Z), there are constants R0,c1,c2 and 1 E0 such that if F =(f,φ):[−T,T]×S → Z is a (J, ν)-holomorphic map with diam F (A0)

2 (5.6) E(F, t) ≤ CE(F, T) ρ 3 (t) ∀|t|≤T 964 ELENY-NICOLETA IONEL AND THOMAS H. PARKER

2/3 ≤ 2/3 ≤ with C =2ρ (T ) 2R0 .IfE(F, T) E0 then there are pointwise bounds of the form

2 2 2 1 (5.7) |df | ≤ c |dF | ≤ c E ρ 3 (t) ∀|t|≤ T. 1 2 0 2

N Proof. When R0 is small the image of F lies in a coordinate ball in Z ×P and we can choose coordinates which identify that ball with the ball B(0,R0) in Cn+1 so that the complex structure Jˆ of (1.3) agrees with the standard structure J0 at the origin. On this ball the Euclidean metric is uniformly equivalent to the metric on Z × PN , so it suﬃces to prove the lemma using the Euclidean norms on Cn+1-valued maps. Writing the (J, ν)-holomorphic map equation (1.3) in terms of the stan- − dard operator ∂ = ∂J0 in those coordinates, we have ∂F =(J J0) dF j. But |J − J0|≤c|F |≤cR0 in our coordinates, giving the pointwise bound

(5.8) |∂F|≤cR0 |dF |. By writing F = u + iv as the sum of its real and imaginary parts, one finds that 4|∂F|2 dt dθ = |dF |2 dt dθ − 2 d(u · dv). Integrating over A =[−t, t] × S1 and using Stokes’ theorem gives 1 2 2 |dF | =2 |∂F| + u · vθ dθ. 2 A A ∂A The boundary term is an integral over two circles. On each, we can replace u − 1 2π byu ˜ = u 2π 0 udθand apply the Hölder and Poincaré inequalities on the circle (5.9) 2 u · vθ dθ = u˜ · vθ dθ ≤ u˜ L2 vθ L2 ≤ u˜θ L2 vθ L2 ≤ |Fθ| .

2 2 2 From the deﬁnition 2∂F = Ft + iFθ and the inequality (a − 2b) ≤ 2a +8b we also obtain 2 2 2 2 2 3|Fθ| =2|Fθ| + |Ft − 2∂F| ≤ 2|dF | +8|∂F| . Together, the previous three displayed equations and (5.8) imply that − 2 2 | |2 ≤ 4 2 2 | |2 1 4c R0 dF 1+4c R0 dF . A 3 ∂A 2 2 Taking R0 small enough that 4c R0 < 1/44 and adding the previous two equations lead to 2 E(t) ≤ E (t). 3 Integrating this diﬀerential inequality from t to T yields (5.6). THE SYMPLECTIC SUM FORMULA 965

On the cylinder [−T,T] × S1, each point lies in a unit disk with Euclidean metric, and F =(f,φ)isJˆ-holomorphic as in (1.3). Standard elliptic estimates then bound |dF |2 at the center point in terms of the energy of F in that unit disk; see [PW, Th. 2.3]. Thus (5.6) implies (5.7).

In the next several sections we repeatedly use the facts that, because ρ2 =2|µ| cosh(2t) is essentially exponential in t, the integrals of its powers in the cylindrical metric satisfy (5.10) α ≤ α −α ≤ −α ρ dt dθ cα ρ0 and ρ dt dθ cα ρ0 for α>0. ρ≤ρ0 ρ≥ρ0 We will also use the bump functions deﬁned as follows. Fix a smooth function β : R → [0, 1] supported on [0, 2] with β ≡ 1on[0, 1]. The function βε(z,w)= β(ρ/ε) has support where ρ2 = |z|2 + |w|2 ≤ 4ε2. When restricted to the curve zw = µ, βε ≡ 1 on the neck region A(ε)={ρ ≤ ε}, and dβε is supported on two annular regions where ε ≤ ρ ≤ 2ε. We can choose β so that its norm in the cylindrical metric (4.5) satisﬁes

(5.11) |dβε|≤2.

We next consider the renormalized maps obtained from fn =(vn,xn,yn) by centering vn and rescaling xn and yn.

Deﬁnition 5.2. In the region ρ ≤ 1 around each node deﬁne renormalized maps fˆ by n ˆ 1 − 1 k − k xn yn fn =(ˆvn, xˆn, yˆn)= vn v¯n,...vn v¯n, s , s az bw i i { | |} wherev ¯n is the average value of vn on the center circle γn = ρ = µn of the neck of Cn.

Whenever λn = xnyn is nonzero xn has no zeros and has (local) winding number s. Hence eachx ˆn has winding number zero, so the functions logx ˆn, and similarly logy ˆn, are well-deﬁned. The convergence (5.5) shows that on | |≥ → x s 1 each set z r we havex ˆn f0 /az =1+O(r)inC , and similarly fory ˆn. Thus there is a constant c so that

(5.12) sup | logx ˆn| + sup | logy ˆn|≤cr ∀n ≥ N = N(r). r≤|z|≤1 r≤|w|≤1

λ Lemma 5.3. For each sequence f as in (5.1) we have lim n = ab at n →∞ s n µn each node.

Proof.ForG = logx ˆ the integral n n 1 2π Gn(ρ)= Gn dθ 2π 0 966 ELENY-NICOLETA IONEL AND THOMAS H. PARKER over the circles with fixed ρ — or equivalently fixed t — satisfies 2π 2π 2π d −1 2π Gn = ∂tGn dθ = (∂t + i∂θ)Gn dθ =2 xn ∂xn dθ. dt 0 0 0

Here xn is x ◦ fn where x is the x-coordinate on Zλ = {xy = λ}. A calculation 2 similar to (4.4) shows that the differential dx : TZλ → C satisfies |dx| = (1 + |λ|2/|x|4)=|x|2/R2 where R2 = |x|2 + |y|2. Also note that νN = O(R)by (1.15a) and that along V , in the coordinates (5.4), J − J0 is acting on normal | N |2 ≤| − N N · vectors is O(R). Hence equation (1.2) gives ∂fn (J J0)df n j + ν 2 2 2 dφn| ≤ cR |dFn| where Fn =(fn,φn). Thus (5.13) −1 2 | −1 ◦ N |2 ≤| |−2 | |2 · 2 | |2 ≤ | |2 xn ∂xn = xn dxn ∂fn xn dxn cR dFn c dFn . | d |≤ 1/3 ≤ These equations and Lemma 5.1 imply that dt Gn c1 ρ . Hence for ρ r and n>N(r) r 1/3 1/3 1/3 (5.14) |Gn(ρ)|≤|Gn(r)| + c1 ρ dt ≤|Gn(r)| + c2 r ≤ c3 r ρ where the last inequality uses (5.12). Since we are free to take r arbitrarily small, this implies that the average of logx ˆn on the center circle γµ satisfies n 1 logx ˆ = G |µ | → 0 2π n n n γµn as n →∞. Symmetrically, the same limit statement holds with G replaced by logy ˆ. For each n we can then integrate the constant λn xnyn log s = log s s = log (ˆxn yˆn) abµn az bw over γµ to see that n λ λ 2π log n = log n = logx ˆ + logy ˆ → 0. abµs abµs n n n γµn n γµn The lemma follows. √ ≤ ≤ Lemma 5.3 allows us to improve (5.12). For each r, µ r 1, the region where ρ = |z|2 + |w|2 is at least r consists of two components, one of the form |z|≥r and one of the form |w|≥r where r is essentially r. On the first, logx ˆn is bounded as in (5.12). In the second region (5.12) gives a bound on logy ˆn, which we can now parlay into a bound on logx ˆn using Lemma 5.3, the equations zw = µn and xnyn = λn, and the last four displayed equations of the above proof. Specifically, in the second region we have | | λn − ≤ | | ≤ 1/3 logx ˆn = log s logy ˆn Gn µn + cr cr . abµn THE SYMPLECTIC SUM FORMULA 967

A similar bound holds for logy ˆn in the ﬁrst region. Thus there is a constant c so that 1/3 (5.15) sup | logx ˆn| + | logy ˆn|≤cr ∀n ≥ N = N(r). r≤ρ≤1 We conclude this section by translating these bounds on the renormalized maps into bounds on weighted Sobolev norms of the original maps fn.

Lemma 5.4. Given a sequence of δ0-ﬂat (J, ν)-holomorphic maps as in (5.3) which satisfy the hypotheses of Lemma 5.1, write fn =(vn,xn,yn) in the V − N ≥ coordinates (5.4) and set fn = vn vn and fn =(xn,yn). Then for each p 2 ≤ 1 ≥ there are constants Cp and N = N(r) so that whenever 0 <δ 3 and n N 1 the restriction of fn to the cylinder A(r)={ρ(t) ≤ r}⊂[−T,T] × S satisﬁes (5.16) |∇ V |p | V |p | 1−s∇ N |p | 1−s N |p −pδ/2 ≤ p/6 fn + fn + ρ fn + ρ fn ρ dtdθ Cp r . A(r)

Proof. Write A(r) as the union of cylinders Ak = {k ≤ t ≤ k +1} of unit size and let ρk be the value of ρ at one end of Ak. Using the Sobolev inequality which bounds oscillation by the L4 norm of the derivative and noting that |df V | = |dv |≤cρ1/3 by (5.7), we have n n (5.17) | V |≤ | |≤ ≤ 1/3 ≤ 1/3 sup fn oscAk vˆn C dvn 4,Ak C ρk Cr A(r) k k k where the last inequality comes from the Riemann sum for ρ1/3dt.Thus |∇ V |p | V |p ≤ p/3 fn + fn cρ pointwise. Integration via (5.10) then gives the first two terms of (5.16). Next, the Calderon-Zygmund inequality of [IS] shows that the Lp norm of G = logx ˆn satisfies dG ≤ C ∂(β G) p,A(r) r p,A(2r) ≤ · −1 C dβr G p,A(2r)\A(r) + xn ∂xn p,A(2r) . We can estimate the last term by integrating using (5.13), (5.7), and (5.10), and can estimate the dβr · G term using (5.11), the bound (5.15) in the region p r ≤ ρ ≤ 2r where dβr = 0, and (5.10). These imply that the L norm of dG is bounded by cr1/3. But then for each cylinder A ⊂ A(r) with unit diameter we can use (5.14) and the Sobolev inequality as in (5.17) to obtain | |≤| | | | sup G avg∂A G + oscA G A ≤ 1/3 ≤ 1/3 ≥ cr + C dG 4,A(r) cr for all n N(r). 1/3 Exponentiating this bound on G shows that |xˆn − 1|≤cr in A(r), and that s in turn gives |dxˆn| = |xˆn dG|≤c |dG|. Consequently xn =ˆxn · az satisfies s+1/3 s (5.18) |xn|≤cρ and |dxn|≤cρ (1 + |dG|) 968 ELENY-NICOLETA IONEL AND THOMAS H. PARKER since |dz/z| is bounded in the cylindrical metric. The same bounds hold for p the yn, so integration, combined with the L bound on dG, gives the remaining part of (5.16).

6. The space of approximate maps

The limit arguments of Section 3 show that as λ → 0 sequences of holomorphic maps fn into Zλ have subsequences which converge to maps into ∪ MV × MV X Y with matching conditions along V , i.e. to maps in s (X) ev s (Y ). The results of Section 5 give further information about the convergence near the matching points: they show that for small λ the maps fn are closely approximated by maps g(z,w)=(v, azs,bws) in local coordinates. Over the next four sections we will reverse this process, showing how one can use MV × MV M s (X) ev s (Y ) to construct a model odels(Zλ) for the space of stable maps into Zλ. The final result is stated as Theorem 10.1. The construction has two main steps. In the first, maps f in a compact set K ⊂MV × MV δ s (X) ev s (Y ) are smoothed in a canonical way to construct maps F into Zλ which are approximately holomorphic. The second step corrects those approximate maps F to make them truly holomorphic. This section describes the canonical smoothing and the resulting space of approximate maps and introduces norms on the space of maps which capture the convergence of the renormalized maps. Those norms lead to a precise statement that the approximate maps are nearly (J, ν)-holomorphic. The maps alone cannot be canonically smoothed — more data are needed. This harks back to the comment at the end of Section 3 that each f will generally be the limit of many maps into Zλ. Recall that an element of MV × MV → s (X) ev s (Y )isamapf : C0 Z0 from a bubble domain C0 whose last (s) nodes xk = yk are mapped into V with contact of order sk. By the construction of Section 4 each such C0 determines an -dimensional family C0(µ), µ =(µ1,...,µ), which smooths those nodes (and leaves other nodes unaffected). Lemma 5.3 indicates that f0 : C0 → Z0 is the limit of maps into Zλ which satisfy

sk (6.1) akbk µk = λ to highest order. That leaves |s| = s1s2 ···s possibilities for µ corresponding to the different choices of root for each µk. As in (5.5), the coefficient ak is the sk-jet of the component of f1 normal to V at xk modulo higher order terms, and so, intrinsically, ∗ ∈ sk ⊗ ak (Txk C) (NX V )f(xk). Globally on the space of relative stable maps there are two complex line bundles ∗ associated with the marked point xk : (i) the bundle evsk NX V obtained by pulling back NX V by the evaluation map at xk and (ii) the relative cotangent THE SYMPLECTIC SUM FORMULA 969 bundle Lk to the domain (over the maps with stable domains, Lk is the pullback ∗ st Lk of the bundle Lk → Mg,n that appears in (4.3)). The leading coefficients in (5.5) are then sections ∗ ∗ ∈ Lsk ⊗ ∈ L sk ⊗ (6.2) ak Γ( k evsk NX V ) and bk Γ(( k) evsk NY V ). ∼ Furthermore, λ is a constant section of NX V ⊗ NY V = C via the trivialization fixed at the beginning of Section 2. Thus (6.1) implies that at each node which is mapped into V the coefficients ak,bk determine a section λ − ∈ L ⊗L sk Γ k k akbk MV × MV th over s (X) ev s (Y ). The sk root of this section is a multisection of L∗ ⊗ L ∗ k ( k) ; considering all k at once defines a multisection of the direct sum L∗ ⊗ L ∗ M of the k ( k) . This gives an intrinsic model for our space odels(Zλ)of approximate maps.

Deﬁnition 6.1 (Model space). For each s and λ = 0, the model space Models(Zλ) is the multisection of L∗ ⊗ L ∗ →MV × MV [ k ( k) ] s (X) ev s (Y ) k=1 whose ﬁber over a map f consists of those µ =(µ1,...,µ) which satisfy

sk λ (6.3) µk = for each k. akbk | | MV × MV This model space is an s -fold cover of s (X) ev s (Y ), and hence its irreducible part is an orbifold for generic V -compatible (J, ν). Elements of the model space are triples (f,C0,µ) where f : C0 → Z0 and µ =(µ1,...,µ) satisﬁes (6.3). Each such element gives rise to an approximate holomorphic map as follows. th From (4.2) and (5.4), we have coordinates (zk,wk) centered on the k node 2 2 xk deﬁned on the region Bk(2) where |zk| +|wk| ≤ 4, and coordinates (v, x, y) ∈ centered on the image f(xk) whenever f(xk) V . Let βµk be the bump func- 1/4 tion (5.11) with ε = |µk| determined from λ by (6.3). Using the notation of sk − x sk − y (5.5), setx ˜(zk)=akzk (1+(1 βµk )h ) andy ˜(wk)=bkwk (1+(1 βµk )h ).

Definition 6.2 (Approximate maps). For each (f,C0,µ) ∈Model s(Zλ), → λ = 0, define an approximate holomorphic map F = Ff,C0,µ : Cµ Zλ by taking Cµ to be the smoothing C0(µ), defining F on Cµ ∩ B (2) by  k  (1 − β )hv(z ), x˜(z ), λ if |z |≥|w | µk k k x˜(z) k k (6.4) F (zk,wk)=  − v λ | |≤| | (1 βµk )h (wk), y˜(w) , y˜(wk) if zk wk , ∈ whenever f(xk) V , and extending by F = f outside the support of k βµk . 970 ELENY-NICOLETA IONEL AND THOMAS H. PARKER

s s Notice that (6.4) is a smooth map into Zλ which is equal to (0,akz k ,bkw k ) | |≥| | in the center region where βµk = 1. In the region where βµk = 0 and zk wk , Cµ is identiﬁed with C1 by (z,w) → (z,0) as in Section 4 and (6.4a) is simply (f1(z),λ) in the chart (2.6a). Symmetrically, (6.4b) reduces to (f2(w),λ)in the chart (2.6b). Hence the maps (6.4) do indeed extend as f over the rest of Cµ.

Remark 6.3. In general, C0 has nodes of two types: those mapped into V and those not mapped into V . When all nodes are of the second type, the gluing problem is relatively easy and has been carefully treated in the literature (see for example [MS] or [RT1]). In fact, the estimates around the two types of nodes are isolated from one another, as follows. Recall that f −1(V ) is a finite set of nodes and, as in Section 4, distinct nodes are separated by distance at least 4. Hence each node x with f(x) ∈/ V lies in a region B = Bk(2) with dist (B,f−1(V )) ≥ 2. Lemma 6.8a below then shows that B is mapped into the complement of a fixed tubular neighborhood of V ⊂ Z; there the geometry of Zλ is uniform in λ and the local estimates involved in the Ruan-Tian gluing apply. Thus near each node xk ∈ C0 which is not mapped into V we can smooth C0 to Cµ, define approximate maps on Cµ ∩ Bk(2) as Ruan-Tian do, and use their estimates on Cµ ∩ Bk(2). That effectively reduces the analysis to the case where there are no such nodes. Bearing that in mind, in the next several sections we will simplify the analysis by assuming that all nodes of C0 lie in f −1(V ), leaving the conflation of estimates to the reader. 6.4 . → Definition (Gluing map) The association (f,C0,µ) (Ff,C0,µ,Cµ) defines a gluing map

(6.5) Γλ : Models(Zλ) → Maps(B,Zλ ×U) were B is the 2-manifold underlying Cµ. The image of Γλ is the space of approximate maps A pproxs(Zλ).

This gluing map is injective for small µ as follows. If Γλ(f,C0,µ)= Γλ(f ,C0,µ) then the (stabilized) curve Cµ = Cµ lies in the family (4.2), C0 = C0 and µ = µ . But then f and f are (J, ν)-holomorphic maps which agree outside the balls Bk(2) and therefore, by the unique continuation property of elliptic equations, agree everywhere. In Section 9 we will show that Γλ is an embedding. Here, as a preliminary, we introduce norms which make the space of maps in (6.5) into a Banach manifold. We will use weighted Sobolev norms tailored for our problem. On the domain we continue to use the cylindrical metric (4.5) and to use (4.10) to THE SYMPLECTIC SUM FORMULA 971 measure distance between curves. In the target, in a neighborhood of V in Z, the tangent bundle TZλ splits as in the proof of Lemma 2.3 as the direct sum ∗ 2 of the tangent space to the complex curve xy = λ in the fiber (L ⊕ L )v = C , which is a complex subbundle Nλ of TZλ, and an orthogonal subspace, which we can identify with TV. Thus in a neighborhood of V each vector field ξ on Zλ decomposes orthogonally as V N (6.6) ξ =(ξ ,ξ ) ∈ Γ(TV ⊕ Nλ). th Fix ρ1 > 0, let Bk(ρ1) denote the unit ball centered on the k node in the family F of (4.2), and let qk ∈ V be the image of that node (we will specify a ∩ precise ρ1 at the end of this section). For small µ the neck Cµ Bk(1) of Cµ is an annulus whose center circle γk is defined by ρ = |µk|. Given ξ, define the average value of the V components on γ k 1 V ∈ (6.7) ξk = ξ Tqk V 2π γk and assemble these averaged vectors at the different nodes into a single vector ∈ ξ =(ξ1,...,ξ) TqV where q =(q1,...,q). We can extend ξk to a vector field (also denoted by ξk) in a neighborhood of qk by parallel translation along geodesics in V emanating from qk and then lifting to a vector field on TZλ V perpendicular to Nλ; that extension has the form (ξk , 0) in the decomposition (6.6). Each ξ then determines a global vector field on Cµ: (6.8) ζ = ξ − βk ξ¯k with zero average value, where βk is the bump function (5.11) with ε = ρ1/2 centered on the kth node. Finally, we fix 0 <δ<1/6 and define a weighted Sobolev norm by p |∇k |p | |p −δp/2 (6.9) ζ k,p,s = ζ + ζ ρ Cµ using the norms, covariant derivative, and volume form associated to the cylindrical metric (4.5) on the domain and the metric induced on Zλ from Z.Thus near each node in the coordinates (5.3) the metric on the domain is the flat cylindrical metric and the weighting function ρ is exponential in t. 6.5. ×U Definition Given a tangent vector (ξ,h) to the space Maps(B,Zλ ) we form the triple (ζ,ξ,h¯ ) as in (6.8) and define the weighted Sobolev m norm

(6.10) ||| (ξ,h)||| m = ζ m,2,s + ζ m,4,s + |ξ¯| + h 01 ∗ where h is given by (4.8). For 1-forms η ∈ Γ(Λ (f TZλ)) we do the same without averaging:

(6.11) ||| η||| m = η m,2,s + η m,4,s. 972 ELENY-NICOLETA IONEL AND THOMAS H. PARKER

1,4 → 0 ||| ¯ ||| By the Sobolev embedding Lloc Cloc the norm (ζ,ξ,k) 1 dominates the C0 norm. Hence we can use it to complete the space of C∞ maps, making ×U Maps(B,Zλ ) a Banach manifold with a Finsler metric given by (6.10).

Remark 6.6. The norms deﬁned above make sense also at λ = 0, where the average value ξ is equal to the value of ξV at the nodes. The choice of norms is guided by the need to dominate the C0 norm and the need to have uniform estimates as λ → 0. Note that as λ → 0 the domain becomes, near each node, isometric to cylinders [−T,T] × S1 with T →∞. Thus the L4 norm does not dominate the L2 norm uniformly in λ, and the L1,4 norm does not give uniform bounds on energy. The choice (6.10) with m = 1 uniformly controls both C0 norms and energy and is also strong enough to initiate the bootstrap argument establishing the regularity of the ﬁxed point at the end of Proposition 9.4.

Each (f0,C0) ∈Kδ can be stabilized as in Remark 1.1 by choosing submanifolds transverse to the image. This procedure stabilizes the domains of all maps in a neighborhood of (f0,C0)inKδ. Because C0 is then stable, we can choose a local chart F of the form (4.2) for a neighborhood of C0 in the universal curve. Elements of Kδ close to (f0,C0) can be regarded as maps f from a fiber of F into Z. That provides a local “chart” which identifies a neighborhood of (f0,C0)inKδ with the set of maps from the fibers of F into Z. In that chart φ = Id in all formulas, and a sequence (fn,Cn) converges if 0 and only if the Cn converge as fibers of F and the fn converge in C , in energy, and in C∞ on compact sets in the complement of the nodes.

Observation 6.7. In the next four sections we will work in the charts just described, seeking estimates which are uniform for (f,C) in the set Kδ of (3.11). Of course, Kδ is compact, so that any estimate which holds locally in a chart at each (f,C) ∈Kδ holds uniformly in Kδ. ∈K Here is one application of this observation. For every (f,C) δ let −1 A(ρ0)= k D(xk,ρ0) denote the region within distance ρ0 > 0off (V )= {xk}. Lemma 6.8. ∈K There are constants ci,ci > 0 such that for every (f,C) δ,

(a) for each ε there is a ρ0 > 0 such that dist (f(A(ρ0)),V) ≤ ε, and for each ρ0 > 0 there is a c>0 such that dist (f(C \ A(ρ0)),V) ≥ c. | |≤ 1/3 ≤ ≤ (b) df c1ρ c2 on C and f C c ρ on some A(ρ0) for each .

(d) The components of (5.5) satisfy |h| + |dh|≤c5ρ on some A(ρ0). THE SYMPLECTIC SUM FORMULA 973

Proof. By Observation 6.7 it suffices to show (a) and (b) on a neighborhood of a fixed (f0,C0) ∈Kδ. If the first assertion of part (a) fails, then there exists an ε0 > 0 and sequences (fn,Cn) of maps and pn of points such that (fn,Cn) converges to (f0,C0) in the stable map topology, pn converges to one of the nodes xk, but the distance from fn(pn)toV is bounded below by ε0. Now, since 0 fn converges in C to f and f(xk) ∈ V this gives a contradiction. Similarly, if the second part of statement (a) fails on every neighborhood of (f0,C0)we can find a sequence (fn,Cn) converging to (f0,C0) and points pn ∈ Cn \ A converging to p ∈ C0 with f(p) ∈ V . Since the pn are uniformly bounded away −1 from the points of fn (V ), p must be a new intersection point, contradicting the invariance of s described in Remark 3.5. Thus (a) holds. Now recall from Section 4 that the special points on any C are separated by distance at least 4. Thus for ρ0 < 1 consider the region B(ρ0) defined as the disjoint union of the ρ0-neighborhoods of the nodes not mapped into V .We can find a ρ0 > 0 small enough so that all maps (f,C) in a small neighborhood of (f0,C0) satisfy the hypotheses of Lemma 5.1 on B(ρ0). That Lemma then 1/3 gives the bound |df |≤c1ρ on B(ρ0) (in cylindrical metric on the domain). The bound f C ≤ c ρ in the cylindrical metric (4.1) is equivalent to f C ≤ c in the the metric gU on the universal curve. Using the latter metric, suppose there is no such bound on C \ B(ρ0) on every neighborhood of (f0,C0). Then there exists (fn,Cn) ∈Kδ converging to (f0,C0) and points pn converging to p ∈ C0 with ρ(pn) ≥ ρ0 and f C →∞. But restricting −1 −1 to fn (X), then to fn (Y ), the sequence pn is bounded away from the nodes, ∞ so stable map convergence implies that fn → f0 in C , a contradiction. The bounds (b) follow after we note that A(ρ0) ⊂ C \ B(ρ0). The coefficients ak and bk, as sections over Kδ as in (6.2), are smooth by Lemma 7.1 below and cannot vanish because of the invariance of s (Remark 3.5). Since Kδ is compact they are uniformly bounded above and below. Fi- nally, when f is expanded as in (5.5) its x component satisfies |f x − azs|≤ | |s+1 s cs z by the C bound in (b). There are similar bounds on other components which, together with the bounds of part (c) yield |h|≤cρ. Essentially similar estimates give |dh|≤c|dz|≤cρ in the cylindrical metric.

We conclude this section by showing that the approximate maps are nearly − holomorphic. The speciﬁc statement is that the quantity ∂JF F νF , which measures the failure of the approximate map to be (J, ν)-holomorphic, is small in the norm (6.11).

Lemma 6.9. There exist constants c and λ0, uniform on Kδ, such that | | ∈K for λ <λ0 and (f,C0) δ each approximate map F = Ff,C0,µ satisﬁes 1 ||| − ||| ≤ | | 2|s| ∂JF F νF 0 c λ . 974 ELENY-NICOLETA IONEL AND THOMAS H. PARKER

Proof. Let A be the union of the annuli Ak in the domain Cµ near the −1 node xk = yk ∈ f (V ) deﬁned by ρ ≤ ρ0 where ρ0 is the constant of Lemma 6.8a for ε small enough to ensure that f(Ak) lies in a geodesic ball in which we have coordinates (v, x, y) of (5.4). First consider the set Cµ \ A. By Lemma 6.8a the image of Cµ \A lies in the region of Zλ with distance at least c to V . In that region the projection gives an identiﬁcation of Zλ with Z0 under which F = f for small λ, while the distance in Z between F (z) ∈ Zλ and f(z) ∈ Z0 is at most c|λ|. That gives the estimate |JF − Jf | + |νF − νf |≤c|λ|. − − We can then bound the quantity ΦF = ∂JF F νF by noting that ∂Jf f νf =0 − ◦ ◦ and ∂JF = ∂Jf +(JF Jf ) dF j:

(6.12) |ΦF |≤(|JF − Jf | + |νF − νf |) |dF |≤c|λ||df | where |df |≤c2 by Lemma 6.8b. Then (5.10) immediately gives 1/p − (6.13) ρ δp/2 |Φ|p ≤ c |λ|. Cµ\A

Now we focus on the half A+ of one Ak where |wk|≤|zk| (the estimates in the other half are symmetric). Omitting the subscripts k, we have local coordinates z,w in which, from (6.4a) and (6.3), − (6.14) F = (1 − β)hv,azs(1 + (1 − β)hx),bws(1+(1− β)hx) 1 and f is given by the same formula with the last entry replaced by zero. Lemma 6.8d shows that, after possibly making ρ0 smaller, we can assume that | x|≤ | − |≤ | s| h 1/2on Ak. We then have F f c bw . Differentiating, noting that w = |µ| exp(t + iθ) satisfies |dw| = |w| in the cylindrical metric on the domain, and using (5.11) and Lemma 6.8cd yields |dF − df |≤c|b| (|dws| + s x s x s 2 2 s |w h dβ| + |w dh |) ≤ c|w |. But |ρw| ≤ 2|µ| in A+ and |µ |≤c|λ| by (6.3) and Lemma 6.8d. Thus s −s (6.15) |F − f| + |dF − df | + |JF − Jf |≤c min{ρ , |λ|ρ } on Ak. Using the bound of Lemma 6.8b we also have |dF |≤|dF − df | + |df | ≤ cρ. Let J0 be the complex structure of the coordinate system on the target used in (6.14); this agrees with J at the origin, so that |JF − J0|≤c|F |≤cρ. As above, the J-holomorphic map equation for f leads to the estimate (6.16) |ΦF |≤|∂0(F − f)| + |(JF − J0) ◦ (dF − df )| + |(JF − Jf ) ◦ dF | + |νF − νf |. s x −1 Also as above, the term |∂0(F − f)|≤|bw ∂0(1 + (1 − β)h ) | is bounded by s s cρ|w |. Furthermore, the one-form ν satisfies |νF − νf |≤c|F − f|≤c|w | in s the the metric on the universal curve, so |νF − νf |≤cρ|w | in the cylindrical metric (4.1). With these facts, (6.16) reduces to 1+s 1−s (6.17) |ΦF |≤c min{ρ , |λ|ρ } THE SYMPLECTIC SUM FORMULA 975 on A+ and by symmetry on all of Ak, with this constant c uniform on Kδ.Now 2 2 2 integrate over Ak using (5.10) and note that ρ = |z| + |w| ≥ 2|zw| =2|µ| | sk |≤ | | ≤|| on Ak. Recalling that δ<1/6, µk c λ , and sk s , and using Lemma 6.8c we obtain 1/p 1 −δp/2 p (2−2sk−δ)/4 | | (6.18) ρ |ΦF | ≤ c |λ||µk| ≤ c |λ| 2 s . Ak The lemma follows by combining (6.13) and (6.18) and summing on k and on p =2, 4.

We can now be precise about the choice of the radius ρ1 of support of the cutoff functions βk which appear in (6.8). Specifically, we take ρ1 to be the ρ0 of the first sentence of the proof of Lemma 6.9. That ensures that for each (f,C0) ∈Kδ the image of the support of βk lies in a region where we have the coordinates (5.4) and where the extension ξk and the vector field (6.8) are well-defined.

7. Linearizations

This section describes the linearization of the (J, ν)-holomorphic map equation as an operator on the Sobolev spaces of Definition 6.5. We describe MV first the moduli space s (X) which defines the relative invariants, then the M MV × MV space s(Z0)= s (X) ev s (Y ) of maps into the singular space Z0. This serves as background for our main purpose: describing the linearization ∗ operator Df,C at an approximate map into Zλ and its adjoint Df,C. These operators are used to be used in Section 9 to correct the approximate maps into holomorphic maps. To begin, consider a V -regular map (f,φ):C → X × U g,n in the space (1.17). For simplicity we usually omit mentioning χ, n and A, and will surpress the φ by considering C to be a curve with the complex structure induced by φ. ∈MV → Thus (f,C) s (X) means that f : C X is a map from a smooth complex −1 curve marked by the (s) points {xk} of f (V ) with corresponding multiplicity vector s =(s1,...s). Our first aim is to describe the linearization (1.10) with the norms (6.10) and (6.11). Thus we decompose ξ into (ζ,ξ) and regard ζ ∗ as an element of the Banach space L1;s,0(f TX) obtained by completing the smooth sections with average value ζ = 0 in the norm ζ 1,2,s + ζ 1,4,s as in (6.10). One can show, as we prove in the context of Lemma 7.3 below, that the linearization extends to a bounded linear map ∗ 01 ∗ (7.1) Df,C : L1;s,0(f TX) ⊕ TqV ⊕ TC Mg,n+ → Ls(Λ (f TX)) defined in terms of the linearization (1.10) by (7.2) Df,C(ζ,ξ,h)=Df,C ζ + βkξk,h 976 ELENY-NICOLETA IONEL AND THOMAS H. PARKER

(the last space in (7.1) is the completion in the norm (6.11) with m = 0). Thus Df,C and Df,C are essentially the same operator; the boldface indicates that we have separated out the average value vector ξ and completed in the weighted norms. Using results from [Loc] it is routine to verify (cf. Lemma 3.4 and Lemma 7.3 below) that for generic 0 <δ<1, (7.1) is Fredholm with respect to these norms. Furthermore, infinitesimal deformations of f : C → X in Ker Df,C preserve the multiplicity vector s =(s1,...,s) (because s can be expressed as a winding number at infinity and the norm (6.9) dominates the C0 norm). MV Thus (7.1) locally models the space s (X). In particular, for generic V - MV ∗ MV compatible (J, ν) the irreducible part s (X) of s (X) is an orbifold of dimension 2 indC Df,C. We also have the following two facts. Lemma 7.1. × MV ∗ → M × (a) st ev : s (X) g,n+ V is a smooth map of Banach manifolds. Lsk ⊗ (b) The leading coefficient (6.2) is a smooth section of the bundle k ∗ MV ∗ evsk NX V over s (X) . Proof. (a) It suffices to show that the linearization is a smooth map MV → everywhere. Let (ξ,h) be a tangent vector to s (X)atf : C X and ∈ decompose ξ = ζ + βkξk with ξk = ξ(xk) Tf(xk)V . The linearization of the map st × ev is (ξ,h) → (h, ξ) which is obviously smooth with our norms. MV (b) Choose a path (ft,Ct)in s and let (ξ,h) be its tangent vector at N s | |s+1 t = 0. Assume for simplicity that = 1 and let ft = atz + O( z ) be the expansion near the single point x with q = f(x) ∈ V . Writing ξ = ζ + βξ with ξ = ξ(x) ∈ TqV and differentiating, we see that the tangential component V N s s+1 ζ (x) ∈ TqV at x vanishes and the normal component is ζ =˙atz +O(|z| ). N Consequently, |a˙ t|≤c|ζ |Cs ≤ c|(ξ,h)|Cs , and that is dominated by c||| (ξ,h)||| 1 by elliptic bootstrapping for the equation Df,C(ξ,h) = 0. Thus the differential of the section (6.2) is bounded.

For maps into the singular space Z0 the linearization is essentially two copies of (7.1), as follows. Regarding f : C0 → Z0 as a pair of maps ∈MV × MV (f1,C1; f2,C2) s (X) ev s (Y ), a variation ξ of f consists of smooth ∗ ∗ sections ξ1 of f1 TX and ξ2 of f2 TY with ξ1(xk)=ξ2(yk) at each node xk = yk which is mapped into V . Thus the domain of the linearization consists of sections (ζ1, ξ1,h1; ζ2, ξ2,h2) with the matching condition ξ1 = ξ2 at the nodes mapped into V , and the linearization extends to a bounded linear map ∗ ⊕ ⊕ M⊕ M→ 01 ∗ (7.3) Df,C0 : L1;s,0(f0 TZ0) TqV TC1 TC2 Ls(Λ (f0 TZ0)). Again, as in Lemma 3.4, one can verify that for generic V -compatible (J, ν) J ∈K in (Z), Coker Df,C0 = 0 at irreducible maps (f,C0). In fact, for f δ this can be done, as in Lemma 4.2 of [IP4], by a perturbation in (J, ν) supported THE SYMPLECTIC SUM FORMULA 977 outside the δ/2 neighborhood of V (maps in Kδ, stabilized as in Observation 6.7 have no components mapped into the δ neighborhood of V ), and hence by perturbations in the class J (Z) of Lemma 2.3. Similarly, the evaluation MV ×MV → × map ev : s s V V is smooth and its image is transverse to the diagonal ∆ for generic (J, ν)inJ (Z). Therefore the irreducible part of the MV × MV −1 space s ev s = ev (∆) is generically a smooth orbifold and its tangent space at an irreducible map (f,C0) is identiﬁed with Ker Df,C0 .

We next turn to the linearization DF at an approximate map F = Ff,C0,µ, which is deﬁned by (7.2) with (f,C) replaced by the approximate map F of Deﬁnition 6.4. Our goal in the remainder of this section is to show that DF extends to a Fredholm operator ∗ ⊕ ⊕ M → 01 ∗ (7.4) DF : L1;s,0(F TZλ) TqV TCµ g,n Ls(Λ (F TZλ)).

We can also consider the formal adjoint of DF with respect to the weighted 2 ∗ L inner products, that is, the operator DF determined by the relation ! " ! " ∗ −δ (ζ,ξ,h), DF η = DF (ζ,ξ,h),η where ζ1,ζ2 = ρ ζ1,ζ2 . Cµ Using (7.2) and (1.11) one ﬁnds that ∗ ∗ − t (7.5) DF η = DF η, Aη, F∗Jη   ∗ δ ∗ −δ  DF η = ρ LF (ρ η) where  −δ V ∇ V  Aη = ρ (∂βk)η + βk Jdfj,η . Cµ k

t ∗ 2 Here F∗ is the adjoint of dF , and LF is the formal L adjoint of the operator LF of (1.11). We will see that this also extends to a Fredholm operator ∗ 01 ∗ → ∗ ⊕ ⊕ M (7.6) DF : L1;s(Λ (F TZλ)) Ls,0(F TZλ) TqV TCµ g,n.

The proof that DF is bounded requires a preliminary lemma which is based on our assumption from Deﬁnition 2.2 that the second fundamental form of V ⊂ Z vanishes.

v x y Lemma 7.2. Let ζ =(ζ ,ζ ,ζ ) be a vector ﬁeld on Zλ in a coordinate chart (5.4). Then along the image of F , ζV =(ζv, 0, 0) satisﬁes |(∇ζV )N |≤ | V | | N V |≤ | | cρ ζ and LF (ζ ) cρ ζ .

Proof. Each coordinate chart (5.4) is foliated by the submanifolds Vx,y of points (v, x, y) with fixed x and y. Those submanifolds are deformations of V = V0,0 so that the assumption of Definition 2.2 implies that the second 2 2 2 fundamental form of Vx,y is O(R) where R = |x| + |y| . In particular, the Vx,y foliate Zλ = {xy = λ} and the second fundamental form of Vx,y in Zλ is 978 ELENY-NICOLETA IONEL AND THOMAS H. PARKER also O(R), which means that|(∇ζV )N |≤cR|ζV |. But we know that R ≤ cρs from Lemma 6.8, and so |(∇ζV )N |≤cρ|ζV |. N V Expanding LF by (1.11), we then see that the first two terms of LF ζ are V dominated by cρ|ζ |. Along V , the remaining terms of LF can be expressed in terms of the second fundamental form h: since νN = 0 along V the term N V (∇ζV ν) is h(ζ ,ν), and since J preserves the normal direction to V along N N N V V , the term (J∇ζV J(X)) = J[(∇ζV (JX)) − J(∇ζV X) ]isJh(ζ ,JX)+ V h(ζ ,X). These terms vanish along V by our assumption hV = 0, and hence | V | N V |≤ | | along Vx,y they are bounded by cR ζ . We conclude that LF (ζ ) cρ ζ after again noting that R ≤ cρs.

Proposition 7.3. ∈K For approximate maps F = Ff,C0,µ with (f,C0) δ the operators (7.4) and (7.6) are bounded uniformly in λ. The index of DF is independent of s and

1 V V indexC D = dim M (X) × M (Y ). F 2 s ev s 1 Proof. By (7.2), (6.8), and (1.11), DF,C(ζ,ξ,h)isLF (ξ)+ 2 JF dF h with ∇ ξ = ζ + βkξk. In the formula (1.11) for LF , the one-forms ν and ν on the domain Cµ are bounded in the metric on the universal curve, so are bounded by cρ in the cylindrical metric (4.1); hence

(7.7) |LF ξ|≤c (|∇ξ| + B|ξ|) with B = |dF | + |ΦF | + cρ.

To proceed, write Cµ as the union of Cµ \ A and A exactly as in the proof of Lemma 6.9. On Cµ \ A there we have the bound |dF | + |ΦF |≤c as in (6.12). On each Ak, |dF |≤cρ from after (6.15), and |ΦF |≤cρ by (6.17). Thus B ≤ cρ globally on Cµ. With that, (7.7) yields the pointwise inequality: | |≤ |∇ | | | | | | | (7.8) DF,C(ζ,ζ,h) c ζ + ρ ζ + LF (βkξk) + JF dF h . k

Next observe that |dβk|≤cρ by the bound (5.11) and the fact, from the last sentence of Section 6, that dβ has support where ρ0/2 ≤ ρ ≤ ρ0. Con- | | | | | | | V | | N | sequently, L(βkξk) = ∂βkξk + βkLξk is less than cρ ξk + L ξk + L ξk |∇ | | | which, in turn, is less than c( ξk + ρ ξk ) after one uses Lemma 7.2 and ∇ the estimate (7.7). Here ξk is the covariant derivative of Zλ, which differs from covariant derivative ∇˜ of the Riemannian metric on Ak. However, as in the proof of Lemma 7.2, ξk is tangent to the submanifold Vx,y whose second | ∇−∇˜ |≤ | |≤ s| | fundamental form is O(R), so that ( )ξk cR ξk cρ ξk . Further- more, because ξk is a constant vector field in the coordinates (5.4) we know |∇˜ |≤ | | that ξk Γk ξk where Γk is a bound for the Christoffel symbols; in fact |Γk(v, x, y)|≤c|(v, x, y)| by the construction of those coordinates (normal coordinates along V extended off V by the exponential map). In particular, at THE SYMPLECTIC SUM FORMULA 979 an image point F (z), Γk is bounded by c|F |≤cρ because |F − f|≤cρ by (6.15) and |f|≤cρ by (5.5) and Lemma 6.8d. Thus | |≤ | | (7.9) LF (βkξk) cρ ξ .

The quantity JF dF h in (7.8) expands to (JF − Jf )dF h + Jf (dF − df )h + v Jf df h . By writing f =(h , x,˜ 0) in the notation of Deﬁnition 6.4 and applying 6.8d, one sees that |Jf df h |≤cρ|h|. Using (6.15) we then have

|JF dF h|≤cρ|h|. With these bounds, the right-hand side of (7.8) simpliﬁes to |∇ζ| + ρ|ζ| + ρ|ξ| + ρ|h|. After integrating as in (6.9), comparing with (6.10), and using (5.10) we obtain 1−δ/2 1−δ/2 (7.10) ||| DF (ζ,ξ,h)||| 0 ≤ c ||| ζ||| 1 + |ξ| + ρ h L2 + ρ h L4 .

Finally, to bound the h terms, consider vectors h =(h0,h1) at a curve C in a ﬁxed chart U ×D as in Deﬁnition 4.3. After lifting to the normalization of −1 C at the nodes of f (V ), the components h0 ∈ T N satisfy elliptic equations which are uniform for C in U. Standard elliptic theory then implies that sup |h0|≤c h0 WP with a constant c uniform on the chart. Adding the similar bound on sup |h1| after (4.6) and comparing with (4.8), we see that sup |h|≤c h . Thus the last two terms of (7.10) are bounded by − − (7.11) |ρh|p ρ δp/2 ≤ c sup |h|p ρp(1 δ/2) ≤ c h p, A A with c uniform on f ∈Kδ by Observation 6.7. The right-hand side of (7.10) is now the norm ||| (ζ,ξ,h)||| 1 of (6.10). We conclude that DF is bounded uniformly in λ = 0 and in f ∈Kδ. The proof for ∗ DF is similar, and the index is given by Lemma 3.4.

8. The eigenvalue estimate

We now come to the key analysis step: obtaining estimates on the linearization DF of the holomorphic map equation along the space of approx- ∗ imate maps. We establish a lower bound for the eigenvalues of DF DF and construct a right inverse PF for DF . In the next section PF will be used to correct approximate maps to true holomorphic maps. To get uniform estimates we fix (J, ν) generic in the sense of Lemma 3.4. We continue to work with δ-flat maps, which we call δ0-flat in this section to avoid confusion with the exponential weight δ of the norm (6.9), which will also K ⊂MV × MV appear. As in (3.11) this δ0 defines a compact set δ0 s (X) ev s (Y ) of (3.11) and corresponding subsets

M δ0 ⊂M A δ0 ⊂A (8.1) odels (Zλ) odels(Zλ) and pproxs (Zλ) pproxs(Zλ) 980 ELENY-NICOLETA IONEL AND THOMAS H. PARKER

M δ0 of the model space and the space of approximate maps. Thus odels (Zλ) K is the inverse image of δ0 under the covering map of Deﬁnition 6.1 and A δ0 M δ0 pproxs (Zλ) is the image of odels (Zλ) under the gluing map (6.5). For the K | | maps (f1,f2)in δ0 Lemma 6.8c implies that λ , deﬁned by (6.3), is uniformly s equivalent to |µk| k for each k. In Sections 8 and 9 we also use the notation (8.2) ∗ ∗ M δ0 ⊂M δ0 A δ0 ⊂A δ0 odels (Zλ) odels (Zλ) and pproxs (Zλ) pproxs (Zλ)

MV × MV for the subsets constructed from the maps in s (X) ev s (Y ) which are irreducible as deﬁned after equation (1.4). Recall from Lemma 3.4 that coker Ds = 0 at irreducible maps and hence the spaces (8.2) are orbifolds. That understood, the aim of this section is to prove the following analytic result.

Proposition 8.1. For each generic (J, ν) ∈J(Z), there is a constant E>0 independent of λ such that the linearization DF at an approximate map ∗ ∈A δ0 (F, Cµ)=Ff,C0,µ pproxs (Zλ) has a right inverse 01 ∗ → ∗ ⊕ M PF : Ls(Λ (F TZλ)) L1;s(F TZλ) TCµ g,n such that

−1 (8.3) E ||| η||| 0 ≤ ||| PF η||| 1 ≤ E ||| η||| 0

∗ Proof. Lemma 8.5 below shows that DF DF is uniformly invertible. There- ∗ ∗ −1 ∗ fore PF = DF (DF DF ) is a right inverse for DF . Since DF and DF are bounded by Proposition 7.3 we have ||| η||| 0 = ||| DF PF η||| 0 ≤ E||| PF η||| 1 and ||| ||| ≤ ||| ∗ −1 ||| ≤ ||| ||| PF η 1 c (DF DF ) η 2 E η 0. → For small λ the domain of an approximate map F = Ff,C0,µ : Cµ Zλ has a neck Nk = Bk(1)∩Cµ around each node of C0. These necks are isometric to cylinders using the metric and coordinates of (4.5). The following lemma 2 ∗ and its corollary give a priori estimates for the formal L adjoint LF of LF on the necks Nk(ε)=Bk(ε) ∩ Cµ.

Proposition 8.2. For δ>0 small there are constants ε0 and c such that ∗ ∈A δ0 for all λ sufficiently small, all approximate maps F pproxs (Zλ) and each 01 ∗ neck N = Nk(ε) with ε ≤ ε0, each η ∈ Ω (F TZλ) satisfies δ |∇ |2 | |2 ≤ δ | ∗ |2 δ |∇ |2 | |2 (8.4) ρ η + η c ρ LF η + c ρ η + η . N N ∂N THE SYMPLECTIC SUM FORMULA 981 δ t δ Proof. Write ρ as the derivative of ψ(t)= 0 ρ (τ) dτ and integrate by parts: ρδ |η|2 = ψ |η|2 dt dθ ≤ |ψ|·2η,∇η + |ψ|·|η|2. N N N ∂N Because ρ2 =2|µ| cosh(2t) satisfies ρ2 ≤ 2|µ|e2t ≤ 2ρ2, we have |ψ|≤cρδ/δ,so 1 δ 2 −2 δ 2 that |ψ|·2η,∇η is bounded by ρ |η| + cδδ ρ |∇η| . Rearranging gives 2 δ 2 δ 2 δ 2 (8.5) ρ |η| ≤ Cδ ρ |∇η| + cδ ρ |η| . N N ∂N

Now on the cylinder N every (0, 1)-form η can be written η = η1 dt − ∗ 2 (Jη1) dθ where η1 is a section of F TZλ. Taking the formal L adjoint of (1.11) and using the bounds |dF |≤cρ, |ν| + |(∇ν)∗|≤cρ from the proof of Proposition 7.3 and the obvious bound |(∇J)∗|

Diﬀerentiating the 1-form ω = η1,J∇tη1 dt + η1,J∇θη1 dθ and moving J past ∇ we also have

∗ dω =(2∇tη1,J∇θη1 + η1, ∇t(J∇θη1) −∇θ(J∇tη1))

≥ 2∇tη1,J∇θη1 + η1,JR(∂t,∂θ)η1−c |∇J||dF ||η||∇η|

≥ 2∇tη1,J∇θη1 + η1,JR(∂t,∂θ)η1−cρ|η||∇η| where R is the curvature of ∇. Combining the last two displayed equations, multiplying by ρδ, integrating by parts, and using the bound 2ρ|η||∇η|≤ ρ|∇η|2 + ρ|η|2 we obtain $ % 1 ∗ (8.6) ρδ|∇η|2 ≤ ρδ |L η|2 + R(∂ ,∂ )η ,Jη 2 F t θ 1 1 N N −d ρδ ∧ ω + ρδω + c ρδ+1(|∇η|2 + |η|2). N ∂N

Because the domain metric is ﬂat, R is the (pulled back) curvature of Zλ. Setting U = F∗∂t and V = F∗∂θ and applying the Gauss equations we obtain Z (8.7) R(U, V )η1,Jη1 = R (U, V )η1,Jη1

−h(η1,V),h(Jη1,U) + h(Jη1,V),h(η1,U) where RZ is the curvature of Z and h is the second fundamental form of Z Zλ ⊂ Z. Since R is bounded, the term containing it is dominated by 982 ELENY-NICOLETA IONEL AND THOMAS H. PARKER c|dF |2|η|2 ≤ cρ2|η|2. Likewise, the necks around any nodes not mapped into V are uniformly bounded away from V by Lemma 6.8a. For those nodes, h is bounded along the image F (Nk). Consequently, the entire right-hand side of (8.7) is bounded by c|η|2 |dF |≤cρ|η|2. For nodes mapped into V , F (Nk) lies in one of the sets Ak used in the proof of Lemma 6.9. In the coordinates (v, x, y)onAk defined by (5.4) Zλ is the set xy = λ and its second fundamental form is h(X, Y )=∇X Y,ν. We can compare h with the second fundamental form h0 of the complex curve 2 xy = λ in C . To calculate h0, note that at each point (x, y) with xy = λ the tangent space T and the normal space N are, respectively, the complex span of the unit tangent vector τ =(x/R, −y/R) and the unit normal vector 2 2 2 ν =(y/R, x/R) where R = x + y . Then h0 is the symmetric complex −3 bilinear map T ⊗C T → N given by h0(τ,τ)=τ · τ,ν ν =2λR ν. Because the Christoffel symbols of the coordinate system (v, x, y) are bounded, the 2-tensor h − h0 on Zλ ∩ Ak is bounded uniformly in λ. Ex- amining the proof Lemma 6.9, we see that the normal component F N of F N N s N — see (6.14) — satisfies |F | + |dF |≤cρ k ≤ cR on Ak. Because ν vanishes along V by condition (1.15a) we have |νN |≤cρ|F N | as in the sentence preceding (6.17); that gives |νN |≤cρR and together with (6.17) shows N that U − JV =(dF + JdFj)(∂θ)=2(ν +ΦF ) satisfies |(U − JV) |≤cρR. It follows that |(h − h0)(F∗v, ·)|≤cρ|v| for any v and that |h0(F∗v, ·)|≤ −3 N c|xy|R |dF ||v|≤c|v|. We can therefore replace h by h0 in (8.7), and then replace V by JU, each time making an error of at most cρ|η|2. The result is

R(U, V )η1,Jη1≤−h0(η1,JU),h0(Jη1,U) 2 +h0(Jη1,JU),h0(η1,U) + cρ|η| .

But h0 is complex linear and J preserves the normal direction, so this reduces to

2 2 2 (8.8) R(∂t,∂θ)η1,Jη1≤−2|h0(η1,U)| + cρ|η| ≤ cρ|η| ; that is, the sign of the curvature is compatible with our inequalities. It remains to bound the ω term in (8.6). As in (4.5) we can introduce i cylindrical coordinates (v ,τ,Θ) on the necks of Zλ by choosing normal coor- { i} dinates v on V centered at f(xk), extending by parallel translation into a neighborhood of V , and writing x = |λ| exp(τ + iθ). Then the metric on Zλ 2 2 2 V 2 2 2 V is gλ = R (dτ + dΘ )+g where R = |x| + |y| =2|λ| cosh(2τ) and g is the metric of V . Direct computations with the formula (6.4) for F show that in these coordinates (i) F∗∂θ = sk∂Θ + O(ρ) and (ii) the Christoﬀel symbols are all bounded and those in the Θ direction are

Θ Θ Θ − τ ΓΘΘ =Γττ =0, ΓΘτ = ΓΘΘ = tanh(2τ). THE SYMPLECTIC SUM FORMULA 983

2 Thus ∇θ = ∂θ + tanh(2τ)J + O(ρ). Recalling that ρ =2|µ| cosh(2t) we also have (8.9) δ δ δ −d(ρ ) ∧ ω = −∂tρ η1,J∇θη1 dt ∧ dθ = δρ tanh(2t)Jη1, ∇θη1 dt ∧ dθ. ∗ Because F gλ is independent of θ in these coordinates, the method of (5.9) and the inequality | tanh(2τ)|≤1 gives the bound 2 − tanh(2τ) Jη1,∂θη1 dθ ≤ |∂θη1| dθ. S1 S1

Replacing ∂θ by ∇θ − tanh(2τ)J + O(ρ) on the right-hand side once gives 2 2 0 ≤ ∇θη1,∂θη1 + cρ |∇η| + |η| S1 S1 and again gives 2 2 2 tanh(2τ) Jη1, ∇θη1 dθ ≤ |∇θη1| + cρ |∇η| + |η| . S1 S1 S1

But tanh(2τ) = tanh(2skt)+O(ρ) and 0 ≤ tanh(2t)/ tanh(2skt) ≤ 1 so that this last inequality and (8.9) yield − d(ρδ) ∧ ω ≤ δ ρδ|∇η|2 + c ρ1+δ(|∇η|2 + |η|2). N N N

Insert this and (8.8) into (8.6) and multiply by the constant Cδ of (8.5). Adding (8.5) and noting that |ω|≤|η|2 + |∇η|2 then gives (8.4) for small δ and ε.

Write ∇δη = ρδ ∇(ρ−δη) where ∇ is as usual the covariant derivative of the cylindrical metric on the domain and the metric induced on Zλ from Z. Note that when δ>0 is small, the L1,2 weighted norm (6.9) deﬁned using ∇δ is equivalent, uniformly in λ, to the one using ∇. Then Proposition 8.2 implies the following:

Corollary 8.3. For δ>0 small there are constants ε0 and c such that ∗ ∈A δ0 for all λ suﬃciently small and all approximate maps F pproxs (Zλ) and 01 ∗ each neck N = Nk(ε) with ε ≤ ε0, each η ∈ Ω (F TZλ) satisﬁes ≤ ∗ (8.10) η 1,2,N c DF η 0,2,N + c η 1,2,∂N ; that is, −δ |∇δ |2 | |2 ≤ −δ | ∗ |2 −δ |∇δ |2 | |2 ρ η + η c ρ DF η + c ρ η + η . N N ∂N

−δ ∗ ∗ Proof. Replace η by ρ η in (8.4) and use (7.5) to replace LF with DF . 984 ELENY-NICOLETA IONEL AND THOMAS H. PARKER

From now on we will fix the weight δ>0 in our norms (6.9) small and generic. We will extend the above estimates on the necks to global estimates, first for the weighted L2 norms, then for the weighted L2 ∩ L4 norms of Defi- nition 6.5.

Lemma 8.4. For each generic (J, ν) ∈J(Z), there is an E>0 such that ∗ ∈A δ0 for all λ suﬃciently small and all F pproxs (Zλ) , the ﬁrst eigenvalue of ∗ DF DF is bounded below by E.

Proof. Suppose the claim is false. Then there are sequences λn → 0, maps → A δ0 Fn : Cn = Cµn Zλn in pproxs (Zλ) and (0, 1) forms ηn along Fn with ∗ → DnDnηn = αnηn and αn 0. Taking the inner product with ηn and taking the adjoint (7.5) using our norms (6.9) we obtain 2 ≥ ∗ 2 (8.11) αn ηn 0,2,s Dnηn 0,2,s on each Cn. We may normalize the ηn so that ηn 1,2,s = 1. By the Bubble Tree Convergence Theorem there is a subsequence of the Fn that converges to ∞ a stable map f0 from C0 = C1 ∪C2 into Z0, and this convergence is in C away from the nodes of C0. On each compact set K in the complement of the nodes, 1,2 the Ls norm (6.9) in the cylindrical metric is uniformly equivalent to the usual L1,2 norm. Standard elliptic theory implies that there is a subsequence ∞ 1,2 of the ηn that converges in C on K to an Ls section η with η 1,2,s ≤ 1 ∗ and D0η = 0 along K (as in the sentence before Observation 6.7, convergence ∞ −1 1 ∞ is C away from nodes). Doing this for the sequence Km = ρ ([ m , )) which exhausts the complement of the nodes and then passing to a diagonal 1,2 subsequence yields a limit η defined on C0 \{nodes} with Ls norm at most ∗ \{ } one and D0η = 0 weakly on C0 nodes . ∗ We next show that D0η = 0 weakly on all of C0; that is, D0X, η =0 ∈ 1,2 ∗ ∈ ∈ M× M for all X =(ζ,ξ,h) where ζ Ls (f0 TZ0), ξ TqV and h TC1 TC2 . Given X and ε>0, we can then choose δ<εso that the norm of X on the region N = ρ ≤ δ satisfies ||| (ζ,0,h)||| 1,N ≤ ε. Then fix a cutoff function β as in (5.11) supported on N with |dβ|≤1 and write X = βX + X1. Since X1 =(1− β)X has support on the outside region where ρ ≥ δ/2wehave ∗ D0X1,η = X1, D0η = 0. On the other hand, βX is (βζ,ξ,βh) because (6.7) is unaffected by the cutoff function. Integrating (7.9) over ρ ≤ δ using (5.10) gives D0(0, ξ,0) 0,2,s ≤ cδ|ξ|≤cε||| X||| 1, and it is easy to see from (6.10) and (5.11) that ||| (βζ,0,βh)||| 1 ≤ 2||| (ζ,0,h)||| 1,N ≤ 2ε. With those facts, Hölder’s inequality, the bound η 0,2,s ≤ 1, and Proposition 7.3 give

|D0(βX),η|≤ D0(βζ,0,βh)+D0(0, ξ,0) 0,2,s · η 0,2,s ≤ cε ||| X||| 1. | | ∗ We conclude that D0X, η = 0, so that D0η = 0 weakly. Pairing against ∗ D0η, which lies in the image of the bounded map (7.6), then shows that ∗ ∗ ∗ D0η,D0η = 0, and therefore D0η =0. THE SYMPLECTIC SUM FORMULA 985

Now for generic (J, ν), as observed after equation (7.3), we have Coker D0 ∗ = 0 . Thus our solution of D0η = 0 must be η = 0. Consequently, for each → ∞ small ﬁxed δ0 we have ηn 0inC on the complement of N = k Nk(δ0) and → − in particular on ∂N. Then ηn 1,2,Cn\N 0 and (8.10) bounds ηn 1,2,N c ηn 1,2,∂N , so that (8.11) and the normalization ηn 1,2,Cn = 1 imply the inequalities

1 1 ∗ ≤ η − ≤ C D η 2 n 1,2,N 4 n n 0,2,N ≤ ∗ ≤ ≤ C Dnηn 0,2,Cn Cαn ηn 0,2,Cn Cαn for large n. That contradicts the assumption that αn → 0, completing the proof.

Lemma 8.5. There is a constant C such that for all λ sufficiently small ∗ ∗ ∈A δ0 ∈ 01 and all F pproxs (Zλ) , each η Ω (F TZλ) satisfies ||| ||| ≤ ||| ∗ ||| η 2 C DF DF η 0 in the norms of Definition 6.5.

Proof. Cover the domain Cµ of F by disks of radius 1 in the cylindrical metric so that each point lies in at most 10 disks. Since ρ varies by a bounded factor across each unit interval in the neck, we can apply the basic elliptic estimate on each disk, multiply by ρ−δ/2 and sum to get ≤ ∗ η 2,p,s c ( DF DF η 0,p,s + η 0,2,s) for a constant c = c(p) independent of λ. Combining the p = 4 and p =2 inequalities we get ||| ||| ≤ ||| ∗ ||| η 2 c ( DF DF η 0 + η 0,2,s) . Using Lemma 8.4 and applying Holder’s inequality for the weighted L2 norm, we obtain 2 ≤ ∗ 2 ∗ c η 0,2,s DF η 0,2,s = η,DF DF η 0,2,s ≤ ∗ ≤ ||| ||| ||| ∗ ||| η 0,2,s DF DF η 0,2,s η 2 DF DF η 0 which combined with the previous inequality give the lemma.

9. The gluing diﬀeomorphism

Recall that the norm (6.10) induces a topology on the space Mapss(B,Zλ ×U). Speciﬁcally, for C0 close maps with the same label s in a chart as in Observation 6.7 we can write (f ,C ) = exp(f,C) (ξ,h) and set (9.1) dist (f,C), (f ,C ) = ||| (ξ,h)||| 1. 986 ELENY-NICOLETA IONEL AND THOMAS H. PARKER

This deﬁnes a distance (the inf of the lengths over all paths piecewise of the above type) and hence a topology on Maps(B,Zλ ×U). Using this distance, we will show that the moduli space of stable maps into Zλ is close to the space of approximate maps, and that these spaces are in fact isotopic. As in Observation 6.7 we work in a chart in which all domains have been stabilized. In that chart we use coordinates normal to the stratum N of -nodal curves in the Deligne-Mumford space as in Section 4. Thus a nearly nodal curve C =(B,j) ∈ Mg,n is written as (C0,µ)=C0(µ) where C0 ∈N and µ =(µ1,...,µ) are coordinates in the normal direction to N. We start by describing a parametrization for a neighborhood of the space A of approximate maps pproxs(Zλ) of (8.1). Consider the Banach space bundle 01 ∗ Λ over Models(Zλ) whose ﬁber at an approximate map F is the completion 01 ∗ of Γ(Λ (F TZλ)) in the weighted norm · 0 of (6.11). We write elements 01 ∈MV × MV of Λ as quadruples (f,C0,µ,η) with (f,C0) s (X) ev s (Y ), µ deter- ∈ 01 ∗ mined from λ by (6.3), and η Λ (F TZλ) with F = F(f,C0,µ). The map 01 → ×U (9.2) Φλ :Λ (ε) Mapss(B,Zλ ), Φλ(f,C0,µ,η) = expF,Cµ (PF η)

(with PF as in Proposition 8.1) is deﬁned on an ε neighborhood of the zero 01 section of Λ and agrees with the gluing map Γλ along the zero section. The following proposition shows that Φλ coordinatizes a neighborhood of A δ ∗ pproxs(Zλ) .

Proposition 9.1. There is a constant c>0 so that, for all small λ,Φλ is a diffeomorphism from an ε-neighborhood of the zero section in Λ01 onto a A δ ∗ ×U neighborhood of pproxs(Zλ) in Mapss(B,Zλ ) that contains at least a cε A δ ∗ neighborhood of pproxs(Zλ) . A A δ ∗ Proof. By Lemma 3.4 the tangent space of s = pproxs(Zλ) at F = → ⊥ Ff,C0,µ : Cµ Zλ has the same dimension as Ker DF = (Im PF ) . In fact, 01 (9.3) TF Λ = TF As ⊕ Im PF because any PF η which lies in TF As satisfies, by (8.3), Lemma 5.3 and Lemma 9.2 below, 1/(8|s|) ||| PF η||| 1 ≤ E ||| η||| 0 = E ||| DF PF η||| 0 ≤ CE |λ| ||| PF η||| 1, so that for small λ, PF η is zero. Next fix a path (ft,Ct,µt)inModels(Zλ) starting at (f0,C0,µ0) and let (ξ,h) ∈ T As be the tangent vector at t = 0 of the corresponding path of 01 approximate maps Ft =Φλ(ft,Ct,µt, 0). Each element τ in the fiber of Λ over (f0,C0,µ0) determines a vector field PF0 τ along the image of F0 in TZλ. After extending τ along F by parallel translation we calculate t d dΦλ (f0,C0,µ0,η0)(ξ, h, τ)= expFt (tPFt τ) = ξ + PF0 τ. dt t=0 THE SYMPLECTIC SUM FORMULA 987

Thus dΦλ is an isomorphism by (9.3). Consequently, Φλ is a local diffeomorphism near the zero section of Λ01. To show injectivity, we suppose that injectivity fails on the disk bundle 01 01 Λ (ε)={η ∈ Λ : η 0 ≤ ε} for every ε. Then for each n there ex- 01 ist elements (fn,C0,n,µn,ηn) =(fn,C0,n,µn,ηn)inΛ (1/n) which have the same image under Φλ. After passing to subsequences, we can assume that { } { } the (fn,C0,n,µn) and (fn,C0,n,µn) converge in the stable map topology → → K ⊂MV × MV to limits f : C0 Z0 and f : C0 Z0 in δ s s . Furthermore, ev → → convergence in the metric (4.10) implies convergence C0,n C and C0,n C to elements on the boundary of the cylindrical end compactification of Mg,n defined at the end of Section 4. These limits C and C consist of the nodal curves C0 and C0 together with, for each, an element of the real torus T . Now fix a compact region R in B which contains no nodes. Then for small → → 1 ||| · ||| λ we have Fn f and Fn f in C on R. Since our 1 norm dominates the C0 norm on maps, the triangle inequality gives lim dist C (µ ),C (µ ) + sup dist (f(x),f (x)) →∞ 0,n n 0,n n n x∈R ≤ c lim (||| PF ηn||| 1 + ||| PF η ||| 1) ≤ c lim (||| ηn||| 0 + ||| η ||| 0)=0 n→∞ n n n n→∞ n by Proposition 8.1. The convergence to zero of the first term implies convergence in the topology of the cylindrical end compactification. Thus (i) C = C, and (ii) f and f agree on R and therefore, as in the argument after (6.5), agree everywhere. Consequently, for large n (fn,C0,n,µn,ηn) and (fn,C0,n,µn,ηn) lie in the region where Φλ is a local diffeomorphism and are therefore equal. That establishes injectivity. Surjectivity onto a cε-neighborhood follows from the first inequality in (8.3).

Lemma 9.2. There are constants C and λ0 such that for 0 < |λ| <λ0, A δ the tangent vectors (ξ,h) to pproxs(Zλ) at an approximate map F = Ff,C0,µ satisfy

1/(8|s|) (9.4) ||| DF (ξ,h)||| 0 ≤ C|λ| ||| (ξ,h)||| 1.

∈A δ Proof. Choose a path Ft = F(ft,C0t,µt) pproxs(Zλ) starting at F with initial tangent vector (ξ,h). Thus Ft are approximate maps constructed from ∈MV × MV the path (ft,C0t) s ev s starting at (f,C0) with initial vector (ξ0,h0). We can regard vector fields on Cµ as living on C0 using the map Cµ → C0 defined by (z,w) → z for |w|≤|z| and (z,w) → w for |z|≤|w| near each node and extended over Cµ as in Section 4. In the notation of Definition 4.3 the variation in the complex structure of Cµ decomposes as h =(h0,h1) while − the variation of C0 is h0 =(h0, 0). In particular, h h0 is the variation h1 of Lemma 4.2, which is supported on B = k Bk, where Bk is the region of 988 ELENY-NICOLETA IONEL AND THOMAS H. PARKER

th 1/4 C0 near the k node where ρ ≤ 2|µk| . With that understood, we will separately estimate the left-hand side of (9.4) on C0 \ B and on each Bk. On C0 \ B we can regard vector ﬁelds on Zλ as living on Z0 by the map (v, x, y) → (v, x) for |y|≤|x|, and (v, x, y) → (v, y) for |x|≤|y|, extended over ∗ Zλ as in (2.6). With these identiﬁcations, DF acts on sections of f TZ0 along 1 C0. Since Df (ξ0,h0) = 0 and DF (ξ,h)=LF (ξ)+ 2 JF dF h by (1.10), we have

(9.5) DF (ξ,h)=[LF (ξ) − Lf (ξ0)]

+[(JF − Jf )dF h + Jf (dF − df )h + Jf df (h − h0)] with h0 = h as above. Under our identiﬁcations we also have Ft = ft in Deﬁnition 6.2, so taking the t-derivative shows that ξ = ξ0. Using (6.15), we have

|DF (ξ,h)|≤|(LF − Lf )ξ| +(|JF − Jf | + |dF − df |) |h| 1−s ≤|(LF − Lf )(ξ)| + c|λ|ρ |h|.

But on C0 \ B the bounds in the proof of Lemma 6.9 (on both Cµ \ A and A) 1 1/4 show that the C distance from Ft to ft is bounded by c|λ| . It follows that 1/4 (LF − Lf )(ξ) is dominated by c|λ| (|∇ξ| + |ξ|). After expanding ξ as in (6.8) |∇ |≤ | | and noting that (βkξk) c ξ by the estimates preceding (7.9), the above bound simpliﬁes to 1/4 1−s |DF (ξ,h)|≤c|λ| |∇(ζ)| + |ζ| + |ξ| + c|λ|ρ |h|.

We can then integrate over C0 \B, bounding sup |h| as in (7.11) and integrating the power of ρ over the sets where ρ ≥ 2|µ|1/4 as in (6.18). The result is ||| ||| ≤ | |1/4 ||| ||| (9.6) DF (ξ,h) 0,C0\B c λ (ξ,h) 1.

Now restrict attention to one Bk and again write ξ as (ζ,ξ). The estimates of Lemma 7.3 show that

(9.7) |DF (ξ,h)|≤|∇ζ| + cρ|ζ| + cρ|ξ| + cρ|h|.

To proceed, we will explicitly find ζ in the region B+ ⊂ Bk where |z|≥|w| by differentiating formula (6.4a). That formula is given in the normal coordinate system (5.4) centered on qt = ft(xk) If we parallel translate the trivialization of Tq0 Z along the path qt, the corresponding coordinate systems are related by diffeomorphisms φt with |φt(v, x, y) − (v + qt,x,y)|≤ct |(v, x, y)| where c depends only on the local geometry of Z near V . Hence on the region where ≤| |1/4 ∗ ρ µk , where β = 1 in (6.4a), we can take the t-derivative of φt Ft at t =0 to obtain s ˙ s ξ =(ξ0, az˙ , bw )(1+O(ρ)) .

In particular, the average value (6.7) is ξ = ξ0, so under the decomposition (6.8) ξ =(ζ,ξ0) where ζ =(0, az˙ s, bw˙ s)(1+O(ρ)) THE SYMPLECTIC SUM FORMULA 989 and therefore |∇ζ|≤cρ(|a˙| + |b˙|). But |a˙| + |b˙|≤c||| (ξ,h)||| 1 by Lemma 7.1b, 1/4 the term ρ|ζ| in (9.7) is bounded by |µk| |ζ| on Bk, and sup |h|≤c h ≤ −δ c||| (ξ,h)||| 1 as in (7.11). We can then multiply (9.7) by ρ and use (5.10) to 1/4 integrate over the region over Bk where ρ ≤|µk| . That yields the bound ||| D (ξ,h)||| ≤ c|µ |1/4 ||| ζ||| + ||| (ξ,h)||| + |ξ| + sup |h| F 0,Bk k 0 1 1−δ/2 1−δ/2 × 2 4 ρ L (Bk) + ρ L (Bk) (1−δ/2)/4 ≤ c|µk| ||| (ξ,h)||| 1

1/|s| with δ ≤ 1/6 and |µk|≤c|λ| . The lemma follows after summing on k and adding (9.8).

V,δ Proposition 9.3. For small δ, ε > 0 there is a λ0 so that Ms (Zλ) lies A δ | | in an ε-neighborhood of pproxs(Zλ) whenever λ <λ0.

Proof. We will prove this by contradiction. Assume that, for some δ>0 → ∈MV,δ there exist ε0 > 0, a sequence λn 0, and (fn,Cn) s (Zλn ) such that A δ the distance from (fn,Cn)to pproxs(Zλn ) is at least ε0 for all n. Then (fn,Cn) has a subsequence which converges as in (5.1) to a limit f0 : C0 → Z0 from an -nodal curve C0. Using coordinates in the normal bundle to the -nodal strata in the Deligne-Mumford space as in (4.2) , we can write each Cn as (C0n,µn,1,...,µn,) where C0n is an -nodal curve close to C0. Choose sk µˆn =(ˆµn,1,...,µˆn,) with λn = akbk (ˆµn,k) for each k; there are |s| choices for eachµ ˆn which differ by roots of unity. The data (f0,C0, µˆn) specifies → A δ approximate maps Fn : C0(ˆµn) Zλn in pproxs(Zλn ) as prescribed in (6.4). We will show that for some choice of the roots of unity a subsequence satisfies

(9.8) dist (C0n(µn),C0(ˆµn) ) + dist(fn,Fn) <ε0 where the left-hand side is the Finsler distance between maps deﬁned in (9.1). Then (9.8) contradicts our assumption and proves the proposition. s Lemma 5.3 shows that (µn/µˆn) → 1 at each node. After passing to a subsequence and modifying our choice ofµ ˆn we have µn/µˆn → 1. But then dist(C0n(µn),C0(ˆµn)) → 0 by (4.10). th Let Ak(ρ0) denote the k neck region {ρ ≤ ρ0} of C0n. Outside the union of the Ak(ρ0) both the approximate maps Fn and the maps fn converge ∞ uniformly to f0 in C , so on this region dist(fn,Fn) → 0asn →∞. Inside each Ak(ρ0), we can write Fn − fn =(ζˆn, ξ¯n) in the notation of (6.7) and 0 (6.8) with ξ¯n → 0 because fn → f0 in C . Write ζˆn = ζn +(Fn − f0) where − ||| ||| ≤ 1/6 | − | ζn = f0 fn. Then ζn 1 cρ0 on Ak(ρ0) by Lemma 5.4, while Fn f0 + |∇(Fn − f0)|≤cρ by (6.15). Integration using (5.10) then gives the bound − ≤ 1−δ/2 Fn f0 1,p,s cpρ0 on the integrals (6.9) over each Ak(ρ0). Combining 990 ELENY-NICOLETA IONEL AND THOMAS H. PARKER these facts and taking ρ0 small enough, we conclude that dist(fn,Fn), which is uniformly equivalent to ||| ζn||| 1 + |ξ| by deﬁnition, is less that ε0/2 for large n. Thus (9.8) holds.

∈A δ ∗ The next step is to correct each approximate map (F, Cµ) pproxs(Zλ) to get a true (J, ν)-holomorphic map (F ,C). Speciﬁcally, (F ,C) will be a solution of the equation

(9.9) ∂jf = νf where (f,C) = expF,Cµ (PF η), where PF is the operator of Proposition 8.1, and where η lies in the Banach 01 ∗ 01 ∗ space Ls(Λ (F TZλ)) obtained by completing Λ (F TZλ) in the norm (6.11) with m =0.

Proposition 9.4. There are constants ε, λ0 and C, uniform on Kδ, such ∈A δ ∗ | | that for each approximate map F pproxs(Zλ) and 0 < λ <λ0 equation 01 ∗ (9.9) has a unique solution η ∈ Ls(Λ (F TZλ)) in the ball ||| η||| 0 ≤ ε, and that 1 | | solution is smooth and satisﬁes ||| η||| 0 ≤ C|λ| 2 s .

Proof. If we write (F ,C ) = exp(F,Cµ)(ζ) where ζ =(ξ,h) then − − (9.10) ∂C F ν(F ,C ) = ∂Cµ F νF + DF ζ + QF (ζ) where DF is the linearization at (F, Cµ) and the quadratic QF satisﬁes (as in [F])

(9.11) ||| QF (ζ1) − QF (ζ2)||| 0 ≤ C ( ||| ζ1||| 1 + ||| ζ2||| 1) · ||| ζ1 − ζ2||| 1

Taking ζ = PF η and noting that DF PF η = η, the equation (9.9) that we must solve becomes, via (9.10), − − (9.12) η + QF (PF η)= ΦF where ΦF = ∂Cµ F νF . 01 ∗ Now deﬁne an operator TF on the Banach space Ls(Λ (F TZλ)) by

TF η = −ΦF − QF (PF η). By (9.11) and (8.3)

||| TF η1 − TF η2||| 0 ≤ C ( ||| PF η1||| 1 + ||| PF η2||| 1) · ||| PF (η1 − η2)||| 1 2 ≤ CE ( ||| η1||| 0 + ||| η2||| 0) · ||| η1 − η2||| 0. 2 Hence whenever ε<1/(4CE ) and ||| TF (0)||| 0 ≤ ε/2, the map TF : B(0,ε) → B(0,ε) is a contraction. Therefore TF has a unique fixed point in B(0,ε), and that fixed point η satisfies (9.12) and ||| η||| 0 ≤ 2||| TF (0)||| 0 =2||| ΦF ||| 0 is ∈ 4 ∈ 1,4 bounded by Lemma 6.9. Finally, since η Lloc we have ζ = PF η Lloc with ∞ DF ζ + QF (ζ)=−PF ΦF ∈ C . Elliptic regularity then shows that ζ and η are smooth. THE SYMPLECTIC SUM FORMULA 991

10. Convolutions and the sum formula for ﬂat maps

We can now assemble the analysis of the previous several sections to show that the approximate moduli space, which is built from maps into Z0,isa good model of the moduli space of stable maps into the symplectic sum Zλ. Recall that in Sections 3 and 5 we showed that as λ → 0 stable maps into Zλ limit to maps into Z0 and that the complex structure on their domains is asymptotically determined by λ and the limit map up to a finite ambiguity s corresponding to the different solutions of the equations µk = λ/akbk. That led to the definition of the model moduli space Models(Zλ) in Section 6. On the other hand, each element of Models(Zλ) defines an approximate holomorphic map by equation (6.4); for each λ that gives a gluing map M ∗ →A ∗ ⊂ ×U Γλ : odels(Zλ) pproxs(Zλ) Maps(B,Zλ ). Proposition 9.4 shows that each such approximate map can be uniquely per- turbed to be a true (J, ν)-holomorphic map. A In this section we will show that pproxs(Zλ) is isotopic to the space MV s (Zλ)ofδ-flat (labeled) maps through an isotopy compatible with the evaluation maps. Thus Models(λ) keeps track of the fundamental homology class MV [ s (Zλ)] which defines the GW and GT invariants of Zλ. Passing to homology, we then define a “convolution” operation and establish a formula of the form V ∗ V (10.1) GTX GTY =GTZ under the assumption that all curves contributing to the invariants are δ-flat along V (this condition will be eliminated in Section 12). We noted in (3.11) that as λ → 0 the limits of the δ-flat maps into Zλ lie in V V the compact set Kδ of M (X)×ev M (Y ). We will work on the corresponding M δ A δ compact sets odels(Zλ) and pproxs(Zλ) defined in (8.1). We will also assume that for generic (J, ν)

(10.2) coker Df,s = 0 for all f ∈Kδ. V V In particular, this is true when the moduli space M (X) ×ev M (Y ) consists of only irreducible maps for generic (J, ν). Theorem 10.1. Fix an ordered sequence s and write |s| = si.For generic (J, ν) for which (10.2) holds and for small |λ|, there is an |s|-fold cover M δ K odels(Zλ) of δ with a diagram M δ −→ MV,δ odels(Zλ) s (Zλ)  Φ1  s  λ s  (10.3) st st M× M −→ξ M 992 ELENY-NICOLETA IONEL AND THOMAS H. PARKER where the top arrow is an embedding and is isotopic to the gluing map (6.5) M δ to odels(Zλ). The diagram commutes up to homotopy. Furthermore, there 1 is a constant c = c(δ) so that the image of Φλ consists of maps which are − | | | | MV (δ c λ )-flat, and the image contains all (δ + c λ )-flat maps in s (Zλ). ∈M δ Proof. For each (f,C,µ) odels(Zλ) the gluing map associates a smooth curve Cµ and an approximate map F = Ff,C,µ : Cµ → Zλ. By Propo- 1 sition 9.1 any map that is Ls close to F =Γλ(f,C,µ) can be uniquely written as (10.4) Φλ(f ,C ,µ,η) = exp (PF η) for F = Ff ,C ,µ , (F ,Cµ ) 0 0,1 ||| ||| for some Ls section η of the bundle Λ with η 0 <ε. Proposition 9.4 shows that for small |λ| there is a unique such η = η(f ,C,µ) such that (10.4) is (J, ν)-holomorphic. Then t Φ (f ,C ,µ,η) = exp tPF η(f ,C ,µ) λ (F ,Cµ ) M δ ×U is a smooth 1-parameter family of maps from odels(Zλ) to Mapss(B,Zλ ) 0 1 − | | MV with Φλ =Γλ and the image of Φλ lying in the (δ c λ )-flat maps in s (Zλ). The uniqueness of η in the fibers of Λ0,1, combined with Proposition 9.1 and 0,1 1 the uniqueness of η in the fibres of Λ imply that Φλ is injective. 1 It remains to show that Φλ is surjective. But Proposition 9.1 shows that A 2δ (10.4) is onto at least a cε neighborhood of pproxs (Zλ) and Proposition 9.3 V,δ implies that Ms (Zλ) lies in that neighborhood when |λ| is small enough. V,δ Hence for small |λ| , each element of Ms (Zλ) can be written in the form δ+c|λ| (10.4) with (f ,C ,µ) ∈Models (Zλ) and η the corresponding fixed point ||| ||| ≤ 1 of Proposition 9.4 satisfying η 0 ε.ThusΦλ is surjective. Diagram (10.3) leads to our first formula expressing the absolute invariants of a symplectic sum Z = Zλ in terms of the relative invariants of X and V Y . Recall that the relative invariant GWX is obtained by forming the space MV χ,n,s(X, A) of relatively stable maps and pushing forward its fundamental homology class by the map MV → M × n ×HV (10.5) εV : χ,n,s(X, A) χ,n X X,A,s. We can also consider the space of stable maps from compact, not necessar- MV ily connected, domains by taking the union of products of χ,n,s(X, A) and again pushing forward in homology. The resulting class in the homology of M × n ×HV χ,n X X,A,s is the relative GT invariant (1.25). As we observed in the introduction (see Figure 1), it is the GT invariant that will appear in the symplectic sum formula. To proceed, we should replace the vertical arrows in Diagram (10.3) by the above maps εV and pass to homology. We will do that in two steps, first THE SYMPLECTIC SUM FORMULA 993

HV n incorporating the spaces X and then including the X . In each case we will see that the operation of gluing maps deﬁnes an extension of the bottom arrow in Diagram (10.3), which we examine in homology.

The convolution operation. We can glue a map f1 into X to a map f2 into Y provided the images meet V at the same points with the same multiplicity. The domains of f1 and f2 glue according to the attaching map ξ of (3.8), while HV the images determine elements of the intersection-homology spaces X,A,s and HV Y,A,s which glue according to the map g of (3.10). The convolution operation records the eﬀect of these gluings at the level of homology. For each s the attaching map (3.8) deﬁnes a bilinear form (ξ)∗ : H∗(M; Q) ⊗ H∗(M; Q) −→ H∗(M; Q) for = (s). Similarly, for each s the map g from (3.10) induces a bilinear form HV ×HV on the homology of Y Y with values in RH2(Z), the (rational) Novikov ring of H2(Z), namely HV Q ⊗ HV Q −→ , : H∗( X ; ) H∗( Y ; ) RH2(Z) × ∩ × h,h s = g∗ h h −1 = g∗[∆A,s (h h )] tA. ε (∆s) A∈H2(Z)

−1 This last equality holds because ε (∆s) is the union of components ∆A,s = −1 −1 ε (∆s) ∩ g (A). Combining the two bilinear forms gives the convolution operator that describes how homology classes of maps combine in the gluing operation.

Definition 10.2. The convolution operator ∗ M×H V Q ⊗ M×H V Q −→ M : H∗( X ; [λ]) H∗( Y ; [λ]) H∗( ; RH2(Z)[λ]) is given by |s| (10.6) (κ ⊗ h) ∗ (κ ⊗ h )= λ2(s) ξ (κ ⊗ κ ) h,h . (s)! (s) ∗ s s The right-hand side of (10.6) includes three numerical factors which keep track of how maps glue when we form the symplectic sum. Recall that the powers of λ record the Euler characteristic in the generating series of the invariants (1.7) and (1.24); the factor λ2(s) in (10.6) reflects the relation (3.7) between the Euler characteristics when we glue along (s) points. The factor |s| is the degree of the covering in Theorem 10.1; this reflects the fact that each stable map into Z0 can be smoothed in |s| = s1 · ...· s ways. Finally, V,δ note that elements in the space Ms (Zλ) in Diagram 10.3 are labeled maps; i.e. they have (s) numbered curves on their domains as explained at the end 994 ELENY-NICOLETA IONEL AND THOMAS H. PARKER of Section 3. But the GW and GT invariants of Zλ are defined using the space of unlabeled stable maps, which is the quotient of the space of labeled maps by the action of the symmetric group. That accounts for the factor 1/(s)! in (10.6). HV HV ∈ Since X is the disjoint union of components X,A,s with A H2(X) and deg s = A · V , there is an isomorphism HV ∼ HV H∗( X ) = H∗( X,A,s) tA. A deg s=A·V ∈ HV Below, we will identify h H∗( X ) with A hAtA, where hA are its compo- HV nents in H∗( X,A). Example 10.3. The formula for the convolution simplifies when there are no rim tori in X and Y , and therefore in Z (cf. (1.19)). Then (i) the relative invariants have an expansion of the form (A.3), (ii) the map g of (3.10) is the s s restriction to the diagonal ∆s ⊂ V ×V , and (iii) the h part of the convolution (10.6) is then given by the cup product with the Poincaré dual of the diagonal: ∗ × ∪ × g h h ∆ =PD(∆s) (h h ). s { p} ∗ s We can then ‘split the diagonal’ by fixing a basis C of H ( s V ) and writing V p × q p × PD (∆s)= Qp,q C C = C Cp p,q p V s { p} V q where Qp,q is the intersection form of V for the basis C and Cp = Qp,q C V { i} ∗ { } is the dual basis (with respect to Q ). If γ is a basis of H (V ), let Cm be ∗ s { i} the basis (A.4) (in the appendix) of H ( s V ) corresponding to γ and let V {Cm∗ } be the one corresponding to the dual basis {γi} (with respect to Q ). The convolution then has the more explicit form (10.7) | | m 2(m) (κ ⊗ h) ∗ (κ ⊗ h )= λ ξ (κ ⊗ κ ) C (h) C ∗ (h ). m! (m) ∗ m m m In passing from s to m, we used the fact that each fixed sequence m corresponds (s) (s)! to = ordered sequences s. (ma,i) m! More generally, let X be a symplectic manifold with two disjoint symplectic submanifolds U and V with real codimension two. Suppose that V is symplectically identified with a submanifold of similar triple (Y,V,W) and that the normal bundles of V ⊂ X and V ⊂ Y have opposite Chern classes. Let (Z, U, W) be the resulting symplectic sum. In this case, (3.10) is replaced by HU,V × HV,W →HU,W (10.8) g : X ε Y Z THE SYMPLECTIC SUM FORMULA 995 which combines with the map ξ(s) to give the convolution operator (10.9) ∗ M×H U,V Q ⊗ M×H V,W Q −→ M×H U,W Q : H∗( X ; [λ]) H∗( Y ; [λ]) H∗( Z ; [λ]) as in (10.6). It describes how homology classes of maps combine in the gluing operation for the symplectic sum. Finally, we include the evaluation maps which record the images of the n marked points. These combine with the projections from (2.7) to give the diagram Model (Z ) s λ s   MV (Z ) ev MV (Z ) s 0 s λ s  s  (10.10) ev (X Y )n ev n (X Y )n (Z )n  λ n  n π π0 0 πλ n (Z0) n which commutes up to homotopy. We can also include the spaces M of V ∗ V curves from Diagram (10.3). Pushing forward then gives π0∗(GTX GTY )= πλ∗(GT(Zλ)) as shown in the next theorem.

Theorem 10.4. Assume that for generic (J, ν) MV (a) all maps in s (Zλ) are δ-ﬂat along V when λ is small, and s MV (b) all maps in s (Z0) are (X, V ) and (Y,V )-admissible as deﬁned in s (1.22).

∗ Then for every α0 ∈ T (Z0) U∪W ∗ U∪V ∗ V ∪W ∗ (10.11) GTZ (π α0)= GTX GTY (π0α0). Note that condition (b) is generically satisﬁed when (X, V ) and (Y,V ) are semipositive.

1 ⊗···⊗ Proof. It suffices to verify this for decomposable elements α0 = α0 n k k k k α0 . Let αV , αX , αY denote the restriction of α0 to V , X and respectively Y . k k We can then choose geometric representatives BV of the Poincaré dual of αV k k k k in V and Poincaré duals BX of αX in X and BY of αY in Y which intersect k ∩ k ∩ k V transversely such that, moreover, BX V = BY V = BV . Then the 996 ELENY-NICOLETA IONEL AND THOMAS H. PARKER

k k inverse image under πλ of B ∪ B gives a continuous family of geometric X k Y BV k ∗ k ∗ representatives Bλ of the Poincar´e dual of πλα0 in H (Zλ). The theorem then follows from Theorem 10.1 by cutting down the moduli spaces on the left of Diagram 10.10 by (BX ,BY ) and the ones on the right by Bλ. Constraints in H∗(M) are handled similarly. The details of such arguments are standard (cf. [RT1]).

We should comment on how the assumption that all maps are δ-flat enters the above proof. Notice that in the statement of Theorem 10.1 the δ-flat maps M δ M in odels(Zλ) are paired with maps in s(Zλ) which are not exactly δ-flat — there is a slight variation in δ. But when all contributing maps are δ-flat, −1 the cut-down moduli space ev (Bλ) ⊂ M(Zλ) limits as λ → 0 to a compact subset of the open set Ms ×ev Ms as in (3.11). Hence for sufficiently small δ the set of elements of the limit set which are δ-flat is the same as the set of 2δ-flat elements, so the variation in δ is inconsequential. Theorem 10.4 is a formula for the GT invariants evaluated on only certain ∗ ∗ constraints in H (Zλ) — those of the form π (α0). The following definition characterizes those constraints. It is based on the diagram induced by the collapsing maps of (2.7) ∗ T (Z0) ∗ ∗ (10.12) π π0 T∗(Z) T∗(X) ⊕ T∗(Y ).

∗ Deﬁnition 10.5. We say that a constraint α ∈ T (Z) separates as (αX ,αY ) ∈ T∗ ∗ ∗ ∈ if there exists an α0 (Z0) so that π α0 = α and π0(α0)=(αX ,αY ) T(H∗(X) ⊕ H∗(Y )).

Here are three observations to help clarify which classes α ∈ H∗(Z) separate. These follow by combining the Mayer-Vietoris sequences for Zλ = (X \ V ) ∪ (Y \ V ):

∗ ∗ ∗ ∗−1 δ ∗ i ∗ ∗ ∗ δ H (SV ) −−−→ H (Z) −−−→ H (X \ V ) ⊕ H (Y \ V ) → H (SV ) −−−→ and the similar one for Z0 with the Gysin sequence for p : SV → V : ∪ ∗ ∗−2 c1 ∗ p ∗ p∗ ∗−1 (10.13) H (V ) −−−→ H (V ) −−−→ H (SV ) −−−→ H (V ).

(a) When the ﬁrst map in (10.13) is injective then all classes α separate. In dimension four, this occurs whenever the normal bundle of V in X is topologically nontrivial.

∗ ∗ (b) In general the separating classes are those α for which j (α) ∈ H (SV ) is in the image of the second map in (10.13). THE SYMPLECTIC SUM FORMULA 997

(c) The decomposition (αX ,αY ), if it exists, is unique only up to elements ∗ ⊕ ∗ ∗−1 → ∗ ⊕ ∗ in the image of δX δY : H (SV ) H (X) H (Y ) (the elements that can be “pushed to either side”).

By Deﬁnition 10.5 and for simplicity when U and W is empty, Theorem 10.4 becomes:

Theorem 10.6. Under the assumptions of Theorem 10.4 suppose that moreover α separates as (αX ,αY ). Then V ∗ V (10.14) GTZ (α)= GTX GTY (αX ,αY ).

Note that when (αX ,αY ) decomposes as α = αX ⊗αY the right-hand side V ∗ V is GTX (αX ) GTY (αY ), but in general (αX ,αY ) is a sum of tensors of the 1 1 ⊗···⊗ k k form (αX + αY ) (αX + αY ) and the right-hand side of (10.14) is the corresponding sum. To focus on the decomposable case we make another deﬁnition: we say α ∗ ∗ is supported oﬀ the neck if the restriction j (α) ∈ H (SV ) vanishes. In that ∗ ∗ case α separates into relative classes αX ∈ H (X, V ) and αY ∈ H (Y,V ), generally in several ways. For each such decomposition Theorem 10.6 gives

V ∗ V (10.15) GTZ (αX ,αY )=GTX (αX ) GTY (αY ).

This was the formula described in [IP3].

Example 10.7. Take α to be the Poincar´e dual of a point in Z. This constraint is supported oﬀ the neck and has two independent decompositions depending on whether the point is in X or Y .

Example 10.8. Suppose α = αX ⊗ αY is supported oﬀ the neck and there are no rim tori in (X, V ) and (Y,V ) and that all curves contributing to the invariants are δ-ﬂat along V . Then we can choose a basis of H∗(V ) and expand the relative GT invariants as in the appendix. Combining (10.15) with (10.7) gives the explicit formula

GTχ,A,Z (αX ,αY ) | | 2(m) m V V = λ GT (αX ; Cm) · GT (Cm∗ ; αY ) . m! χ1,A1,X χ2,A2,Y A=A1+A2 m − χ1+χ2 2(m)=χ

Note that from the deﬁnition of relative invariants, the only terms contributing are those for which A1 · V = (m)=A2 · V . E. Getzler has pointed out that the formula above can be neatly expressed in terms of the generating series 998 ELENY-NICOLETA IONEL AND THOMAS H. PARKER

(A.6) and the intersection matrix QV of V , speciﬁcally,

GTZ (αX,αY )   2 V ∂ ∂  V V = exp aλ Q GT (αX )(z) · GT (αY )(w) . ij ∂z ∂w X Y a,i,j a,i a,j z=w=0

Because the decomposition of separating constraints α is not unique, we can often choose several diﬀerent decompositions, and use Theorem 10.14 to get several expressions for the same GT invariant. That yields relations among relative GT invariants. In Section 15 we will use that idea to derive recursive formulas which determine the relative invariants in some interesting cases.

11. The space PV and the S-matrix

1 Starting from the normal bundle NX V of V in X, we can form the P bundle PV = P(NX V ⊕ C)

over V by projectivizing the sum of the normal bundle NX V and the trivial complex line bundle. Let π : PV → V be the projection map. In PV , the zero section V0 and the infinity section V∞ are disjoint symplectic submanifolds, both symplectomorphic to V . Moreover, note that PV # PV = PV . V0=V∞ Under the natural identification of V0 with V∞, the convolution operation V∞,V0 (10.9) defines an algebra structure on H∗(M×H ; Q[λ]). This allows us PV to multiply by GT invariants. Of particular interest are the invariants with no constraints on the image, that is GTV∞,V0 (α) with α = 1, which give an PV operator V∞,V0 V∞ V (11.1) GT (1) ∗ : H∗(M×H ; Q[λ]) → H∗(M×H 0 ; Q[λ]) PV PV PV defined by a power series as in (1.24). This operator is key to the general symplectic sum formula given in the next section. In this section we describe (11.1) and its inverse and develop some examples. Each (J, ν)-holomorphic bubble map f into PV projects to a map fV = π ◦ f into V . Although fV may not be (J, ν)-holomorphic, we can still ask whether fV is stable, using the second definition of stability given before (1.4); namely, f is stable if its restriction to each unstable domain component is nontrivial in homology.

Deﬁnition 11.1. A(V∞,V0)-stable map f : C → PV is PV -trivial if each of its components is an unstable rational curve whose image represents a multiple of the ﬁber F of PV . THE SYMPLECTIC SUM FORMULA 999

Thus the PV -trivial curves are rational curves representing dF with one marked point on the zero section and one on the inﬁnity section, both intersect- V∞,V ing with multiplicity d. Let MI denote the set of P -trivial maps in M 0 V PV and consider the disjoint union

V∞,V (11.2) M 0 = MI M PV R where MR is the set of non-PV -trivial maps. For the next lemma we ﬁx a compatible triple (ω, J, g)onPV for which the projection π : PV → V is holomorphic and is a Riemannian submersion; such a triple is described before the proof of Proposition 6.6 of [IP4]. Then each J-holomorphic map (f,j)intoPV projects to a J-holomorphic map (π ◦ f,j) into V .

Lemma 11.2. (a) MI is both open and closed. The corresponding decomposition of (11.1) is

V∞,V ∞ (11.3) GT 0 (1) = I + RV ,V0 ; PV that is, the PV -trivial maps contribute the identity to the GT invariant.

(b) The non-PV -trivial maps have E(fV ) ≥ αV , where αV is the constant of Definition 3.1. (c) For each fixed A, n and χ, the corresponding term in the convolution Rm = R ∗···∗R vanishes for m large enough. Therefore, the inverse of GT is well defined by: ∞ −1 (11.4) GTV∞,V0 (1) = (−1)mRm. PV m=0

Proof. (a) Clearly MI is closed. To show that the complement of MI is closed, suppose that a sequence (fi) in the complement converges to a trivial map f in the topology of the space of stable maps. Then the homology classes converge and so, after passing to a subsequence, we can assume that each fi represents dF . Similarly, the stabilizations of the domains converge in the Deligne-Mumford space, so we can assume that all domain components of each fi are unstable. But then the fi lie in MI. We conclude that MI is both open and closed. Finally, the decomposition (11.2) gives splitting (11.3) of the GT invariant because convolution by elements of MI is the identity.

(b) If E(fV ) <αV then, as in the proof of Lemma 1.5b of [IP4], every component of the domain is unstable and fV is trivial in homology and therefore f represents a multiple of F . (c) For each (J, ν) , we shall bound the number N for which there are maps in the moduli space defining the convolution RN . That moduli space consists of maps f from a domain C (whose Euler class χ and number n of 1000 ELENY-NICOLETA IONEL AND THOMAS H. PARKER marked points is fixed) to the singular manifold PV # ···#PV obtained from P N copies of V by identifying the infinity section of one with the zero section of the next. Furthermore, these f decompose as f = f j where f j is a map th from some of the components of C into the j copy of PV . j Fixing such an f, let N1 be the number of f whose domain has at least one stable component Cj. These components appear in the stabilization st(C). But st(C) lies in the space Mχ,n of stable curves, and hence has at most dim Mχ,n components. This gives an explicit bound for N1 in terms of χ and n. j The remaining N2 = N −N1 of the f each have unstable domain Cj with j j π∗[f (Cj)] ∈ H2(V ) nontrivial, and so satisfy E(π ◦ f ) >αV by (b) above. j Furthermore, since Cj is unstable f is (J, 0)-holomorphic. We therefore have j N2αV ≤ E(π ◦ f ) ≤ E(π ◦ f) ≤ CωV ,π∗A + Cν, where the first sum is over those j contributing to N2 and the last inequality is as in the proof of Lemma 12.1 below. Since the right-hand side depends only on A, χ, and (J, ν), this bounds N2 and hence N.

Deﬁnition 11.3. The S-matrix is deﬁned to be the inverse of the GT invariant of Lemma 11.2: −1 S = GTV∞,V0 (1) . V PV

(Note that this depends not just on V but on the normal bundle to V and the 1-jet of (J, ν) along V .)

The symplectic sum of (X, U, V ) and (PV ,V∞,V0) along V = V∞ is a symplectic deformation of (X, U, V ), and so has the same GT invariant. The convolution then deﬁnes an operation

U,V V∞,V0 U,V H∗(M×H ; Q[λ]) ⊗ H∗(M×H ; Q[λ]) −→ H∗(M×H ; Q[λ]). X PV X ∗ Thus for each choice of constraints α ∈ T (PV ,V∞ ∪ V0), the GT invariant of PV relative to its zero and infinity section defines an endomorphism V∞,V0 U,V (11.5) GT (α) ∈ End H∗(M×H ; Q[λ]) PV X which describes how families of curves on X are modified — “scattered”— as they pass through a neck modeled on (PV ,V∞,V0) containing the constraints α. The identity endomorphism in (11.5) is always realized as the convolution by the element

V∞,V0 I ∈ H∗(M×H ; Q[λ]) PV corresponding to that part of GT coming from PV -trivial maps. THE SYMPLECTIC SUM FORMULA 1001

1 Example 11.4. When V = P , PV → V is one of the rational ruled surfaces with its standard symplectic structure. If we wish to count all pseudo- holomorphic maps, without constraints on the genus or the induced complex structure, the relevant S-matrix is the relative GT invariant with (κ, α)= (1, 1). This case works out neatly: Lemma 14.6 below implies that SV = Id.

Example 11.5. When we put no constraints on either the domain or the image S is an operator given in terms of GTV∞,V0 by the S-matrix expansion V PV (11.2). In cases where there are no rim tori in PV , we can expand the GT invariants in the power series (A.6) of the appendix. Let GTχ,A(Cm; Cm ) denote the relative invariant of F satisfying the contact constraints Cm along V∞ and Cm along V0. Then the S-matrix SV has an expansion like (A.6) with coeﬃcients

V∞,V0 S (C ; C )=δ − GT (C ; C ) χ,A m m m,m PV ,χ,A m m | | 2(m1) m1 V∞,V0 + λ GTP ,χ ,A (Cm; Cm1 ) m1! V 1 1 A1+A2=A m1 − χ1+χ2 2(s1)=χ

V∞,V0 ×GT (C ∗ ; C ) − ... . PV ,χ2,A2 m1 m

12. The general sum formula

In all of our work thus far we have assumed that the (J, ν)-holomorphic maps we are gluing are δ-flat as in Definition 3.1. In this section we remove this flatness assumption and prove the symplectic sum formula in the general case. The idea is to reduce the general case to the δ-flat case by degenerating along many parallel copies of V . Thus instead of viewing Zλ as the symplectic sum X #V Y along V we regard it as the symplectic sum of 2N + 2 spaces: X and Y at the ends and 2N middle pieces each of which is a copy of the ruled space PV associated to V — see Figure 2 of the introduction. The pigeon-hole principle then implies that for large N all holomorphic maps into Zλ are close to maps which are δ-flat along each ‘seam’ of the 2N-fold sum.

Lemma 12.1. There is a constant E = Eχ,n,A(J, ν) such that every (J, ν)- holomorphic map into Z representing a class A ∈ H2(Z) has energy at most E.

Proof. In an orthonormal frame {e1,e2 = je1} on the domain, the holomorphic map equation is f∗e1 + Jf∗e2 =2ν(e1). Taking the norm squared and ∗ 2 2 ∗ noting that f∗e1,Jf∗e2 = f ω(e1,e2) give |df | =2|ν| +2f ω(e1,e2). The energy of (f,C) is therefore the L2 norm of ν plus the topological quantity ω, A plus a term — the symplectic area of C in the universal curve — which depends only on χ and n. The lemma follows. 1002 ELENY-NICOLETA IONEL AND THOMAS H. PARKER

For the remainder of this section we ﬁx the data χ, n, A which determined the constant E of Lemma 12.1 and ﬁx an integer N with

(12.1) NαV >E where αV < 1 is the constant of Deﬁnition 3.1. § Fixing a small λ0, we partition the neck (2.5) of Zλ0 (see Figure 3 of 2) into 2N segments Zj using the coordinate t from (2.4): j { ∈ | − − ≤ ≤ − } Z = z Zλ0 (j N 1) Nt(z) (j N) ,j=1,...,2N.

Let Z0 be the singular symplectic manifold obtained by symplectically cutting − − 1 Zλ0 at the values tj(z)=j N 2 in the middle of each of these segments. The construction of Section 2 deﬁnes a family

(12.2) Z→D ⊂ C2N of smoothings of Z0 as in Theorem 2.1 but with 2N necks. The fiber over (µ1,...,µ2N ), defined for |µ||λ|, is a space Z(µ1,...,µ2N ) with a neck of j size µj inside each Z . The fiber Z0 over µ = 0 is the singular space obtained by connecting X to Y through a series of 2N − 1 copies of the rational ruled P manifold V associated with V . We symplectically identify Zλ0 with a fixed generic fiber Z(µ0) as depicted in Figure 2 of the introduction. ≤ ε M M Fix δ>0 such that δ 10N and consider the space = χ,n,A(Z(µ0)) j of holomorphic maps into Z(µ0). Let f denote the restriction of f ∈Mto f −1(Zj). We can then define an open cover of M that keeps track of the values of j for which the energy Eδ(f j)ontheδ neck around the cut is small as in equation (3.4). Specifically, to each subset {i1,...,ik} of {1,...,2N} we associate the open subset of M i1,...ik δ j (12.3) M = f ∈M|E (f ) <αV /2 for j = i1,...,ik .

i ,...i Lemma 12.2. The M 1 k cover M = Mχ,n,A(Z(µ0),A) and set theo- retically (12.4) M = Mi − Mi1,i2 + Mi1,i2,i3 − ... .

∈M j ≤ Proof. Each f has j E(f ) E(f)

Now every f ∈Mi1,...ik has small energy in the segment Zj for j = i1,...,ik. Replacing µj by λµj for those values of j (and keeping the remaining µj fixed) defines a 1-parameter subfamily Zλ of (12.2). That family degenerates in the middle of exactly k of the segments Zj. At each of those degenerations f is δ-flat in the sense of Definition 3.1. Hence as λ → 0

V∞,V − (12.5) Mi1,...ik →MV × (M 0 )k 1 × MV . X ev PV ev Y

Since the closure of Mi1,...ik is a compact subset of the set of δ/2-flat maps, we can apply the sum formula (10.6). For fixed χ, n and A we obtain 2N − − 2N k 1 V ∗ − k 1 V∞,V0 ∗ V (12.6) GTX#Y =GTX ( 1) GTP GTY . k V k=1 This formula appears to be dependent on the number of cuts 2N. However, there is a way to rewrite it to see that it is independent of N. Note that after multiplication by GT the middle sum is a binomial expansion; in fact, by Lemma 11.2c, 2N 2N 2N − 2N − 1 − (1 − GT) 1 − (−R) − (−1)k 1 (GT)k 1 = = =GT1. k GT GT k=1 Thus the middle part of (12.6) is exactly the S-matrix of Definition (11.3). This gives the symplectic sum formula in the general case.

Theorem 12.3 (symplectic sum formula). Let (Z, U, W) be the symplectic sum of (X, U, V ) and (Y,V,W) along V . Suppose that α ∈ T∗(Z) is supported oﬀ the neck as in Example 10.8. For any ﬁxed decomposition (αX ,αY ) of α the relative GT invariant of Z is given in terms of the invariants of (X, U, V ) and (Y,V,W) and the S-matrix (11.3) by U,W U,V ∗ ∗ V,W (12.7) GTZ (α)=GTX (αX ) SV GTY (αY ).

Implicit in this is the assumption that the right-hand side of (12.7) is well-defined. As in Section 10, that will be the case if (X, V ) and (Y,V ) are semipositive or more generally if the condition (b) in Theorem 10.4 is generically satisfied. As noted in Remark 1.1 these assumptions are probably not essential. Theorem 12.3 holds more generally when α separates as in Definition 10.5, except that the definition of the S-matrix needs to be enlarged. Instead V∞,V0 T∗ of restricting GTP to α = 1 we restrict it to the subtensor algebra V of ∗ V T (PV ) generated by the kernel of the composition

∗ ∗ i ∗ p∗ ∗ H (PV ) −→ H (SV0) −→ H (V0) 1004 ELENY-NICOLETA IONEL AND THOMAS H. PARKER where SV0 is the circle bundle of the normal bundle to the zero section V0 in PV , p∗ is the integration along its ﬁber and i : SV0 → PV is the inclusion. In that case we get an S-matrix deﬁned by V0,V∞ −1 (12.8) SV =(GTP ) V T∗ V In the important case when U and W are empty Theorem 12.3 expresses the absolute invariant of Z in terms of the relative invariants of X and Y .

Theorem 12.4. Let Z be the symplectic sum of (X, V ) and (Y,V ) and ∗ suppose that α ∈ T (Z) separates as (αX ,αY ) as in Deﬁnition 10.5. Then V ∗ ∗ V (12.9) GTZ (α) = (GTX SV GTY )(αX ,αY ). where SV is the S-matrix (12.8).

If moreover α decomposes as α = αX ⊗ αY then (12.9) becomes V ∗ ∗ V GTZ (α)=GTX (αX ) SV (αV ) GTY (αY ) ∈ T∗ P where αV V is the pullback to V of the restriction of α to V . As a check, it is interesting to verify the symplectic sum formula in one very simple case where the GW invariant is simply the Euler characteristic.

Example 12.5. Fix an elliptic curve C with one marked point (and fixed complex structure). Consider the (J, ν)-holomorphic maps with domain C and representing the class 0. When ν = 0 all such maps are maps to a single point, so the moduli space is X itself. Furthermore, the fiber of the obstruction bundle at a constant map p is H1(C, p∗TX), which is naturally identified with TpX. The (virtual) moduli space for ν = 0 consists of the zeros of the generic → section ν = C ν of this obstruction bundle TX X. Thus this particular GW invariant is χ(X). Similarly, when ν = 0 the moduli space of V -regular curves is X \ V and its V -stable compactification, defined in [IP4], is X. To compute the GW invariant relative to V , we need to know how many of these point maps become V -regular after we perturb to a generic V -compatible ν = 0. Because any V -compatible ν is tangent to V along V the corresponding section ν has χ(X) zeros on X, and of those, χ(V ) lie on V . Thus the relative invariant V − is GWX = χ(X) χ(V ). Moreover, the only contribution of the S-matrix in this case comes from the relative invariant of (PV ,V0 V∞) which counts maps with fixed domain C representing the class 0. But similarly, this invariant is GWV0,V∞ = χ(P )−2χ(V ) = 0, and therefore the S matrix does not contribute PV V in this case. The symplectic sum formula then reduces to the formula

χ(X)+χ(Y ) − 2χ(V )=χ(X#V Y ). Much more interesting examples will be given in Section 15. THE SYMPLECTIC SUM FORMULA 1005

Finally, one can also include ψ and τ classes as constraints. As in [IP4], 2 φi ∈ H (Mg,n) is the ﬁrst Chern class of Li, the relative cotangent bundle at th the i marked point pi. There is a similar bundle Li over the space of stable maps whose ﬁber at a map f is the cotangent space to the (unstabilized) domain curve, and whose Chern class is denoted by ψi. It is also useful to ∗ pair each ψi class with an αi ∈ H (Z) and consider the ‘descendent’ τk(αi)= ∗ ∪ k evsi (αi) ψi . It is a straightforward exercise, left to the reader, to incorporate these constraints into Theorems 12.3 and 12.4.

13. Constraints passing through the neck

Not every constraint class α ∈ H∗(Z) separates as in Definition 10.5. Yet for applications it is useful to have a version of the symplectic sum formula for more general constraints — ones whose Poincaré dual cuts across the neck. ∗ Since the Poincaré dual of α ∈ H (Z) restricts to a class in H∗(X, V ) such a general symplectic sum formula will necessarily involve relative GT invariants of classes α ∈ H∗(X \ V ). That requires generalizing the relative invariant V ∗ GTX , which was defined in [IP4] only for constraints in H (X). We begin by recalling the ‘symplectic compactification’ of X \ V which was used in [IP4]. Let Xˆ be the manifold obtained from X \ V by attaching as boundary a copy of the unit circle bundle p : SV → V of the normal bundle of V in X, and let p : Xˆ → X be the natural projection. Suppose that Z is a symplectic sum obtained by gluing Xˆ to a similar manifold Yˆ along SV . We can then consider stable maps in Z constrained by classes B in Hk(Z), i.e. the set of stable maps f with the image f(x) of a marked point lying on a geometric representative of B. If we restrict g to the Xˆ side, such geometric representatives define constraints associated with classes in H∗(X,∂ˆ Xˆ). Specifically, given a class B ∈ H∗(X,∂ˆ Xˆ), we can find a pseudo-manifold P with boundary Q and a map φ : P → X so that φ(Q) ⊂ ∂Xˆ represents B and use this to cut-down the moduli space. Thus for generic (J, ν) MV ∩ εV s (X, A) p(φ(P ))

deﬁnes an orbifold with boundary denoted by

V (13.1) GTX,A,s(φ).

After we cut down by further constraints of the appropriate dimension, this reduces to a finite set of points, giving numerical invariants constructed using φ. This is particularly simple when B ∈ H∗(X \ V ), i.e. when B can be represented by a map into Xˆ \ ∂Xˆ. The cobordism argument of Theorem 8.1 of [IP4] then shows that the relative invariants (13.1) are well-defined. Note that these relative invariants depend on B ∈ H∗(X \ V ), not on its inclusion 1006 ELENY-NICOLETA IONEL AND THOMAS H. PARKER i∗B ∈ H∗(X). For example, rim tori and the zero class in H2(X,∂ˆ Xˆ) have the same image under p : Xˆ → X, but might have different invariants (13.1). In general the constrained invariant (13.1) will not be well-defined but will depend on the choice of φ. The space (13.2) J V × Maps((P, Q), (X,∂ˆ Xˆ)) has a subset n there is a V -stable (J, ν)-holomorphic W = (J, ν, φ) | map f with f(pi) ∈ p(φ(Q)) ⊂ V i=1 where for some map the marked point pi lands on the projection of φ(Q)intoV . In general, W will have real codimension one, and thus will form walls which separate (13.2) into chambers.

Lemma 13.1. The number (13.1) is constant within a chamber. When B =[φ] satisﬁes p∗[∂B]=0then there is only one chamber, and therefore (13.1) depends only on B.

Proof. Any two pairs (f,φ) that lie in the same chamber can be connected by a path (ft,φt) with ft(pi) ∈ φt(P \Q). The cobordism argument of Theorem 8.1 of [IP4] then proves the ﬁrst statement. Each B in the kernel of p∗∂ can be represented by a map φ as above with φ(Q) of the form p−1(R) for some k − 2 cycle R in V . After restricting the last factor of (13.2) to such φ, the wall W has codimension two, giving the second statement.

The following lemma relates the invariants associated with diﬀerent chambers.

Lemma 13.2. (a) Two maps φ1,φ2 : P → X that agree on ∂P deﬁne a ∗ class a =[φ1#(−φ2)] ∈ H∗(X \V ) with Poincare´ dual α = PD(a) ∈ H (X, V ); the corresponding invariants are related by V V V (13.3) GTX (φ1)=GTX (φ2)+GTX (α). (b) If φ1,φ2 deﬁne the same class in H∗(X, V ) then there exists φ : R → ∂Xˆ with ∂R = Q1 (−Q2) such that φ agrees with φ1 on Q1 and agrees with φ2 on Q2. Then φ1 and φ2#φ have the same boundary. Moreover, V V V ∗ VV (13.4) GTX (φ2#φ )=GTX (φ2)+GTX GTF (φ )

This actually means that in order to extend the deﬁnition of the relative invariants from [IP4], we only need to pick one geometric representative B (any one) such that [B] ∈ H∗(X, V ), [∂B]=β for each β ∈ Ker [H∗−1(SV ) → H∗−1(X)]. THE SYMPLECTIC SUM FORMULA 1007

Altogether, the invariants can be thought of as giving a map

V T∗ \ −→ M×H V (13.5) GTX : (X V ) H∗( X ) which is noncanonical: it depends on the actual representatives for the class α as described in Lemma 13.2. With this extended deﬁnition of the relative invariants, the proof of The- orem 10.4 carries through. That proof began by choosing geometric representatives of constraints α which separate. For a general constraint α ∈ H∗(Z) we can still choose a geometric representative B of the Poincar´e dual, and consider its restrictions BX and BY to (X,∂ˆ Xˆ) and (Y,∂ˆ Yˆ ) respectively. The remainder of the proof still applies, giving a sum formula relating the invariants GTZ (α)ofZ to the relative GT invariants (13.5) of X and Y cut down by the constraints BX and BY .

14. Relative GW invariants in simple cases

The symplectic sum formula of Corollary 12.4 expresses the invariants of X#Y in terms of the relative invariants of X and Y . In the next section we will apply that formula to spaces that can be decomposed as symplectic sums where the spaces on one or both sides are simple enough that their relative invariants are computable. That strategy can succeed only if one has a collection of simple spaces with known relative invariants. This section provides four families of such simple spaces. In some of the examples below the set R of rim tori is nontrivial. In those V cases we will give formulas for the invariants GWX defined in the appendix although, as the examples will show, it is sometimes possible to compute the V GWX themselves even though there are rim tori present. 14.1. Riemann surfaces. For Riemann surfaces one can consider the GW invariants as absolute invariants or relative to a finite set of points. These invariants count coverings, and the homology class A is simply the degree d of the covering. In dimension two the symplectic sum is the same as the ordinary connect sum — one joins two Riemann surfaces by identifying a point on one with a point on another, and then smoothing. Of course, to apply the sum formula one must first find SV , which in this case is built from the relative invariants of 1 (P ,V) where V = {p0,p∞} two distinct points and where all the constraints lie on V . In that context, we fix a nonzero degree d and two sequences s, s that describe the multiplicities of points at the preimages of p0 and p∞ respectively.

Lemma 14.1. V The invariants GWd,g,s,s with no constraints except those on V = {p0,p∞} vanish except when g =0and s and s are single points with 1008 ELENY-NICOLETA IONEL AND THOMAS H. PARKER multiplicity d. In that case

V GWd,0,s,s =1/d

Moreover, in dimension two the S-matrix is always the identity.

Proof. This invariant is the oriented count of the 0-dimensional compo- MV nents of d,g,s,s . But by (1.21) MV − − − dimC d,g,s,s =2d +2g 2+(s) deg s + (s ) deg s =2g − 2+(s)+(s ) is zero only if g = 0 and (s)=(s) = 1, i.e. s and s specify single points with multiplicity d. If we stabilize, there is only one such map, given by the → d V P equation z z , so its contribution to GWd,0,s,s is 1/d. This map is -trivial, and hence contributes as the identity to the S-matrix. The same dimension count gives the invariant with one constraint:

Lemma 14.2. ≥ V When d 2, the invariants GWd,g,s,s (b) of maps with one simple branch point over a ﬁxed point in the target P1 and no other constraints except those on V = {p0,p∞} vanish except when g =0and (s)+(s )=3, V in which case GWd,0,s,s (b)=1.

Perhaps the most interesting two-dimensional example is the g = 1 invariant of the torus T 2.

Lemma 14.3. Fix an integer k ≥ 0. Then the g =1invariants of the torus relative to a set V of k points form a series V 2 V 2 d GW1 (T )= GWd,1(T ) t d that is equal to the generating function for the sum of the divisors σ(d)= l, l|d namely ∞ ∞ ltl (14.1) G(t)= σ(d) td = . 1 − tl d=1 l=1

Proof. This is a matter of counting the (unbranched) covers of the torus. That was done in [IP1] for k = 0. In general, for each degree d cover, each point of V has d inverse images, each with multiplicity one. Following the notation of [IP4] we order the inverse images and divide by d!, leaving us with G(t) again. THE SYMPLECTIC SUM FORMULA 1009

14.2. T 2 × S2. Next we consider the g = 1 invariants of X = T 2 × S2. Thinking of this as an elliptic fibration over S2, we fix a section S = {pt}×S2 and two disjoint fibers F and denote the corresponding homology classes by s and f. Focusing on the classes df and s + df for d ≥ 0, we can form generating functions for the absolute GW invariants and the GW invariants relative to one or two copies of the fiber. F First consider the classes df , where the invariants GWdf , 1,GWdf , 1, and F,F \ GWdf , 1 have dimension 0 by (1.21). There are no rim tori in X F , and when V is one or two copies of the fiber we have df · V = 0, so that = 0 and V is a F F,F point in (1.20). Therefore GWdf , 1 has values in H2(X) and GWdf , 1 has values V in H = H2(X) ×R. Thus all three invariants can be written as power series with numerical coefficients.

Lemma 14.4. The genus one invariants GW and GWF in the classes df are given by d F d GWdf , 1 tf =2G(tf ) and GWdf , 1 tf = G(tf ) d d with G(t) as in (14.1). The corresponding relative invariants GWF,F are indexed by classes df + R for rim tori R and these all vanish: F,F GWdf , 1 tdf +R =0. d

Proof. The generic complex structure on a topologically trivial line bundle over T 2 admits no nonzero holomorphic sections. After projectivizing, we get a complex structure on T 2 × S2 for which the only holomorphic curves representing df are multiple covers of the zero section F0 and the inﬁnity section F∞. This is a generic V -compatible structure for V = F0 or F0 ∪ F∞. As in Lemma 14.3 these contribute G(t) to the power series for these invariants. (Note that for the relative invariant, we compute only the contribution of curves that have no components in V ).

The invariants for the classes s + df are more complicated. By (1.21) the corresponding moduli spaces have real dimension 4, and so become points in HV X after imposing two point constraints; these constraints can be either points p ∈ X \ V ,orC1(q), a contact of order 1 to V at a ﬁxed point q ∈ V . Again rim tori R appear only for the invariant relative to two copies of a ﬁber.

Lemma 14.5. The genus one invariants GW and GWF in the classes s + df , d>0, are 2 d F d GWs+df , 1(p ) t =2tG (t) and GWs+df , 1(p; C1(q)) t = tG (t). d d 1010 ELENY-NICOLETA IONEL AND THOMAS H. PARKER

The corresponding relative invariants GWF,F can be indexed by classes s + df + R for rim tori R and those with two point constraints on V vanish:   2tG (t) if β = p, p and R =0, F,F GWs+df +R,1 (β) ts+df +R =  tG (t) if β = p; C1(q) and R =0, d 0 if β = C1(q0); C1(q∞).

2 2 Proof. We can compute using the product structure J0 on T × S . Con- sider a J0-holomorphic map representing s+df , passing through generic points p1 and p2, whose domain is a genus 1 curve C = ∪Ci. The projection onto the second factor gives a degree 1 map C → S2,soC must have a rational component C0 which represents s. The projection of the remaining components is zero in homology, therefore they are multiple covers of the fibers. Because the total genus is one there is only one such component. In summary, for the product structure J0, the only g = 1 holomorphic curves representing s + df have two irreducible components, one of them a section S, and the other a multiple cover of a fiber F/∈ V . A simple computation 1 shows that H (C, TX|C ) = 0 for these curves, so that J0 is generic for these classes. The constraints require that S pass through p1 and F pass through p2, or vice versa. For each of those two cases there are d choices of the marked point on the domain of F , so the count is the same as in Lemma 14.4 with G(t) replaced by tG(t). This gives the first formula. The count for the second formula is similar. Any V -regular genus 1 holomorphic map through an interior point p and a point q ∈ V has two components: a section through q and a d-fold cover of a fiber F through p. The fiber domain can be marked in d ways, giving the count tG(t). For the invariant relative two copies of the fiber F , there are rim tori, but the discussion above implies that for J0 the only holomorphic curves in the classes s+df +R appear only for R = 0 (where these curves define what R =0 means).

14.3. Rational ruled surfaces. Here let Fn be the rational ruled surface whose fiber F , zero section S and infinity section E define homology classes with S2 = −E2 = n. We will compute some of the relative invariants GWV with V = S ∪E and with no constraint on the complex structure of the domain (κ = 1). Fix a nonzero class A = aS + bF and two sequences s, s of multiplicities that describe the intersection with S and E respectively. The relative GW invariant with no constraint on the complex structure and k marked points lies in the homology of Xk × S × E with = (s) and = (s). After ∗ imposing constraints α =(α1,...,αk) ∈ T (X), we have S,E ∈ ⊗ GWA,g,s,s (α) H∗(S ) H∗(E ) THE SYMPLECTIC SUM FORMULA 1011

∼ ∼ 1 where S = E = P . Noting that the canonical class of Fn is K = −2S+(n−2)F and deg s = E · A = b and deg s = S · A = b + na,wehave S,E − dimC GWA,g,s,s (α)=(n +2)a +2b + g 1 −(deg s − (s)) − (deg s − (s )) + k − deg α =2a + g − 1+ + + k − deg α.

But S × E has complex dimension + , so the pushforward of the moduli space represents zero in homology unless dimC Mg,k,s,s (Fn,A) ≤ + .Thus the invariant vanishes unless (14.2) 2a + g ≤ 1 + deg α − k. Lemma 14.6. S,E The invariants GWA,g,s,s with no constraints except those on V = S ∪ E vanish except when A = bF , g =0, and s and s are single points with multiplicity b>0. In that case

S,E 1 GW = (S ⊗ 1+1⊗ E) . bF,0,s,s b

Moreover, the S-matrix in Fn is the identity.

Proof. It suffices to show that the only contributions to GW from classes A = aS + bF come from unstable rational domains with a = 0, i.e. from Fn-trivial maps. Taking κ = α = 1, (14.2) implies that A = bF and g = 0 or 1. Moreover, because every bF curve intersects both E and S, we have + ≥ 2, and when g = 0, stability of the domain requires that + ≥ 3. In these cases MV F ≥ the moduli space g,s,s ( n,bF) is either empty or has dimension 2. Suppose that the moduli space is nonempty and the above stability conditions hold. Since E and S are copies of P1, H∗(S) ⊗ H∗(E ) is generated by point or [P1] constraints. Then for each generic (J, ν) there are maps f in the moduli space whose images passes through at least two fixed points p, q ∈ E∪S in generic position. Take (J, ν) → (J0, 0) where J0 is a complex structure with 1 a holomorphic projection π : Fn → P . In the limit we obtain a connected stable mapf0 through p and q with components representing aiS + bif such that bF = aiS + bif. But then each ai = 0, so that the image of π ◦ f0 is a single point containing π(p) and π(q). This cannot happen for generic p, q. MV F F Thus g,s,s ( n,bF) consists of n-trivial maps (cf. Definition 11.1) representing A = bF . Such maps contribute the identity to the S-matrix.

Lemma 14.7. ∈ F \ ∪ V Fix a point p n V with V = E S. Then GWA,g,s,s (p) vanishes except in the following cases:

S,E (i) GWbF,0,s,s (p)=1when s and s are single points with multiplicity b>0.

S,E × (ii) GWS+bF,0,s,s (p)=SVs SVs whenever deg s = b, deg s = b + n. 1012 ELENY-NICOLETA IONEL AND THOMAS H. PARKER

V ≤ Proof. From (14.2) we have GWaS+bF,g,s,s (p) = 0 unless 2a+g 2. Thus either (i) a = 0, or (ii) a = 1 and g =0. In case (i) each map contributing to the invariant represents bF , passes MV F − through p, and hits E and S. Hence dim g,s,s ( n,bF)=g 1+(s)+(s ) with (s)+(s) ≥ 2. The limiting argument used in Lemma 14.6 then shows V that GWbF,g,s,s (p) vanishes unless g = 0 and (s)=(s )=1.Thuss and s are single points of multiplicity b, and the maps pass through p. Moving to the fibered complex structure, one sees that there is a unique such stable map for each b>0. This gives (i). MV F In case (ii) the moduli space 0,s,s ( n,S+bF ) has dimension (s)+(s ) ≥ MV F and is empty unless b 0. That means that the image of 0,s,s ( n,S+bF )isa multiple of SVs × SVs , so that the invariant vanishes except when all contact points on E and S are fixed. By the adjunction inequality, any irreducible curve C representing S + bF is rational and embedded, so we can compute the P 0 F O invariant by intersections in (H ( n, Fn (S + bF )) (the standard complex 1 structure on Fn is generic for these curves C because h (C; O(S + bF )|C )= 1 1 0 h (P ; O(n +2b)) = 0). But h (Fn, O(S + bF )) = n +2+2b, and each of the conditions imposed (including multiplicities) are linear conditions. Thus the number of curves representing S + bF passing through a point p and meeting E and S at fixed contact points is 1.

14.4. The rational elliptic surface. As a final example we consider the rational elliptic surface E. Let f and s denote, respectively, the homology classes of a fiber and a fixed section of an elliptic fibration E → P1. The following lemma describes the invariants relative to a fixed fiber F in the classes A = s + df where d is an integer. In this case there are rim tori in E \ F , suggesting that one use the summed invariant GW defined in the appendix. However, the lemma shows that the sum contains only one nonzero term (as happened in the last case of Lemma 14.5).

Lemma 14.8. The genus g relative and absolute invariants of E in the classes s + df ∈ H2(E) are related by:

g F g F g GWs+df , g (p )=GWs+df , g (p ; C1(f)) = GWs+df , g (p ; C1(f)) where the second equality means that GWF can be indexed by classes s+df +R for rim tori R and these vanish whenever R =0 .

Proof. The ﬁrst equality holds because generically all maps contributing to the absolute invariant are V -regular. That is true because if some component of a stable map is taken into V = F , then that component must have genus at least 1. But then the remaining components have genus less than g, so cannot pass through g generic points. THE SYMPLECTIC SUM FORMULA 1013

The second equality follows from a projection argument like the one used for Lemma 14.5. Consider a curve C = ∪Ci representing s + df which is holomorphic for a fibered complex structure J0 on E. Since the projection 1 to P gives a degree one composition, C must have a rational component C0 that intersects each fiber in exactly one point, while the other components are multiple covers of fibers and so represent df ∈ H2(E \ F ). Moreover, C0 is an 2 embedded section representing s. Since s = −1 then C0 must be the unique holomorphic curve in the class s. Thus the only curves in the class s + df + R appear only for R =0. The invariants of Lemma 14.8 will be explicitly computed in Section 15.3.

14.5. Rational relative invariants. Counting rational curves requires only the g = 0 relative invariants and the corresponding S-matrix. The following two propositions show that these are particularly simple: the S-matrix is the identity and the relative invariant is the same as the absolute invariant in the absence of rim tori.

Proposition 14.9. When g =0, s =(1,...,1) and A ∈ H2(X), the relative invariant (summed over rim tori as in (A.1)) equals the absolute invariant: 1 V GW (α; C ([V ])) = GW (α) (s)! A,0 s A,0 n where α =(α1,...,αn) ∈ (H∗(X)) .

Proof. Fix a generic V -compatible pair (J, ν). Recall that (J, ν) is generic for curves that have no components in V , and also its restriction to V gives a generic pair on V . However, for a curve entirely contained in V , even though (J, ν) is generic when the curve is considered in V , it might not be generic when the curve is considered in X. For any genus g and ordered sequence s, consider the natural inclusion: MV →M g,n,s(X, A) g,n+(X, A) where A ∈ H2(X) and where the left-hand moduli space involves a union over rim tori. When s =(1,...,1), any element in Mg,n(X, A) that has no · MV components in V lifts in (A V )! ways to an element of g,n,s(X, A). We will show that for generic V -compatible (J, ν), when g = 0 the contribution of the moduli space of curves with some components in V to the absolute invariant vanishes, and therefore the two invariants are equal. For simplicity, start with the case when f has only one component, and this is entirely contained in V . Then (s)=A · V = c1(NX V ) · A and the moduli space of such curves has M − · − − dim V,A0,g = 2KV A0 + (dim V 6)(1 g) MV − = dim X,A,g +2g 2 1014 ELENY-NICOLETA IONEL AND THOMAS H. PARKER as in equation (6.4) of [IP4]. This means that for genus g = 0 the dimension of the moduli space of curves entirely contained in V is two less than the (virtual) dimension when considered as curves in X. Therefore if the virtual dimension in X is 0, there are no curves in V which could contribute. The general case of a curve with some components in V and some oﬀ V follows similarly.

Proposition 14.10. The g =0part of the S-matrix is the identity for any V and any normal bundle N.

Proof. By (11.3) this statement is equivalent to showing that there is no contribution to the g = 0 GW-invariant coming from maps into PV which are not P -trivial. Consider the 0 dimensional moduli space MV0,V∞ (γ) V PV ,A,0,s constrained only along V0 and V∞, such that the corresponding GW invariant is not zero. By Theorem 1.6 of [IP4] the same moduli space would be nonempty for the submersive structure associated with a generic ν on V (as deﬁned before ∈M M Lemma 11.3). Then each f R would project to a map fV in V,π∗A,0,s that passes through the γ constraints. But counting virtual dimensions using equation (6.4) of [IP4], we see that

M MV∞,V0 − N − dim ∗ (γ) = dim (γ) 2index D =2g 2 V,π A,0,s PV ,A,0,s s

is negative when g = 0, so this moduli space is empty for generic νV .

15. Applications of the sum formula

This last section presents three applications of the sum formula: (a) the Caporaso-Harris formula for the number of nodal curves in P2, (b) the formula for the Hurwitz numbers counting branched covers of P1, and (c) the formula for the number of rational curves representing a primitive homology class in the rational elliptic surface. These formulas have all recently been established using Gromov-Witten invariants in some guise. Here we show that all three follow rather easily from the symplectic sum formula.

15.1. The Caporaso-Harris formula. Our first application is a derivation of the Caporaso-Harris recursion formula for the number N d,δ(α, β) of curves in P2 of degree d with δ nodes, having a contact with a line L of order k at αk fixed points, and at βk moving points, for k =1, 2,... and passing through the appropriate number r of generic fixed points in the complement of L. For this we consider the pair (P,L), which can be written as a symplectic connect sum:

2 2 (15.1) (P ,L)#(P1,E,L)=(P ,L) L=E THE SYMPLECTIC SUM FORMULA 1015 where (P1,E,L) is the rational ruled surface with Euler class one with its zero section L and its inﬁnity section E. We can then get a recursive formula for the GT invariant of (P2,L) by moving one point constraint pt to the P side, and then using the symplectic sum formula. The splitting (15.1) is along a sphere V = E = L, so there are no rim tori. The relative invariant therefore lies in the homology of SV and is invariant under the action of the subgroup of the symmetric group that switches the order of points of the same multiplicity. A basis for this homology is given by 1 1 (A.4), where {γi} with γ = p a point and γ2 =[P ] is a basis of H∗(V ). To recover the notation of Caporaso and Harris, for each sequence (ma,i), denote αa = ma,1 and βa = ma,2, and let α =(α1,α2,...), β =(β1,β2,...). | | | | αi Then m! and m correspond to α!= i αi and α = i i . With this change of coordinates,

d,δ 1 L r N (α, β)= GT P2 (p ,C ) m! χ,dL, m − − − where χ 2δ = d(d 3) is the “embedded Euler characteristic”, r =3d + g − 1 − αi − (βj − 1), and we are imposing no constraints on the complex structure of the curves. Similarly, let

a,b,χ 1 E,L N (α ,β ; p; α, β)= GT (C ; p; C ) m!m! χ,aL+bF,P m m denote the number of curves of Euler characteristic χ in P representing aL+bF that have contact described by (α,β) along E,(α, β) along L and pass through an extra point p ∈ P (we prefer to label these numbers using χ rather then the number of nodes). By Lemma 14.6 the S-matrix vanishes. The symplectic sum theorem then implies: − N d,χ(α, β)= |α |·|β |·N d ,χ (α ,β ) · N d d ,b,χ (β ,α; p; α, β) where the sum is over all α, β and all decompositions of (dL, χ)into(dL, χ) and ((d − d)L + bF, χ) such that χ = χ + χ − 2(α) − 2(β). Combining Lemmas 14.6 and 14.7 we see that there are exactly two types of curves that E,L P contribute to the relative GW invariant GTP1 (Cm; p; Cm )of with one ﬁxed point p.

(1) Several g = 0 unstable domain multiple covers of the ﬁber, one of them say of multiplicity k passing through the point p, corresponding to the situation d = d and

β = β + εk; α = α − εk

where εk is the sequence that has a 1 in position k and 0 everywhere else. 1016 ELENY-NICOLETA IONEL AND THOMAS H. PARKER

(2) Several g = 0 unstable domain multiple covers of the ﬁber together with one g = 0 curve in the class L + aF passing through p and having all contact points with E and L ﬁxed say described by α0 and α0; this corresponds to d = d − 1 and the situation ≥ ≥ α = α0 + α ; β = α0 + β; equivalently β β; α α

In each situation above, the number of V -stable curves is 1. In the second α β case, note that there are α choices of α0 and β of α0. Moreover, for each P-trivial curve its invariant combines with its corresponding multiplicities in |m|(m)! to give 1. Therefore, the remaining multiplicity in case 1 is k, while | | | − | in case 2, it is α0 = β β . Putting all these together, we get: α β − N d,δ(α, β)= kNd,δ (α − ε ,β+ ε )+ |β − β| N d 1,δ (α ,β ) k k α β where the last sum is over all β ≥ β, α ≥ α. This is exactly the Caporaso- Harris formula.

15.2. Hurwitz numbers. The method of Section 15.1. can also be applied for maps into P1. In that case the symplectic sum formula yields the cut and paste formula for Hurwitz numbers that was first proved using combinatorics by Goulden, Jackson and Vainstein in [GJV] (recently Li-Zhao-Zheng [LZZ] have derived a similar formula using [LR]). The Hurwitz number Nd,g(α) counts the number of genus g, degree d covers of P1 that have the branching pattern over a fixed point p ∈ P1 specified by the unordered partition α of d, while the remaining branch points are simple and located at fixed points in the target P1. We can get at these numbers by regarding the pair (P1,p) as a symplectic sum:

(15.2) (P1,p)=(P1,x)#(P1,y,p). x=y

We then get a recursive formula for the GW invariant of (P1,p) by moving one simple branch point b to the (P1,y,p) side and applying the symplectic sum formula. In fact the Hurwitz numbers are the coefficients, in a specific basis, of the GW invariants of P1 relative to a point V = p. More precisely, each unordered partition α =(α1,α2,...)ofd defines numbers ma =#{i | αi = a}; let Cm be ∗ the corresponding basis (A.4) (in this case the basis {γi} of H (V ) has only one element). Then

p r Nd,g(α)=GWP1,d,g(b ; Cm) is the number of degree d, genus g covers that have the branching pattern over 1 p ∈ P determined by α, and r =2d − 2+2g − 2 − (a − 1)ma other ﬁxed, THE SYMPLECTIC SUM FORMULA 1017 distinct branch points in the target. (Note that the branching order is the order of contact to p = V .) The corresponding generating function (A.6) is

ma r p p r (za) u d 2g−2 F =GW1 = GW 1 (b ; C ) t λ . P P ,d,g m m ! r! a a Now apply the symplectic sum formula to the decomposition (15.2), putting r − 1 branch points on the ﬁrst copy of P1 and one on the second copy. Since there are no rim tori and the S-matrix vanishes by Lemma 14.1 we obtain p |m | p − p,p r · r 1 · (15.3) GW (b ; Cm)= GT (b ; Cm ) GT (Cm ; b; Cm) d,g m ! d,χ1 d,χ2 − where the sum is over all m =(m1,m2,...) and all χ1,χ2 such that 2 2g = χ1 + χ2 − 2(m ) and so that the attached domain is connected. But GT = exp GW and Lemma 14.2 implies that the only possibility for the last factor in (15.3) is a union of trivial spheres together with a degree a sphere constrained by Ca at one end and Ci,j with i + j = a at the other end (plus the branch point in the middle). Therefore there are only two possibilities for the other factor and for the partition α corresponding to m.

(1) α =(i, j, β) and α =(i + j, β) for some i, j, β, so the covering map has genus g and degree d.

(2) α =(a, β) and α =(i, j, β) with a = i + j. Then χ1 =2g − 4 so the covering map is either genus g − 1 and degree d or genus g1,g2 of degrees d1, d2.

The sum formula (15.3) can then be written as a relation for the generating function, namely 1 $ % ∂ F = ijλ2z ∂ ∂ F + ∂ F · ∂ F +(i + j)z z ∂ F . u 2 i+j zi zj zi zj i j zi+j i,j≥1 This is the ‘cut-join’ operator equation DF = 0 of [GJV]. It clearly determines the Hurwitz numbers recursively. The same formula works to give the ‘Hurwitz numbers’ counting branched covers of higher genus curves.

15.3. Curves in the rational elliptic surface. We next consider the invariants of the rational elliptic surface E → P1. Using the notation of Section 14.4, we focus on the classes A = s + df where d is an integer. The numerical g invariants GWA,g(p ) then count the number of connected genus g stable maps in the class s+df through g generic points (with no constraints on the complex structure of the domain). For each g these define power series g d Fg(t)= GWd,g(p ) t ts d≥0 1018 ELENY-NICOLETA IONEL AND THOMAS H. PARKER where t = tf . Recently, Bryan-Leung [BL] proved that $ % g (15.4) Fg(t)=F0(t) G (t) with G as in (14.1) and 12 1 (15.5) F (t)=t . 0 s 1 − td d As mentioned in the introduction, this formula is related to the work of Yau- Zaslow [YZ] and to more general conjectures (such as those stated in [Go]) about counts of nodal curves in complex surfaces. We will use our symplectic sum theorem to give a short proof of this formula, beginning with the g = 0 case. The proof is accomplished by relating F (t) to the similar series of elliptic (g = 1) invariants ∗ d H(t)= GWd,1(τ1[f ]) t ts d≥0 ∗ 2 ∗ where f ∈ H (E) is the Poincaré dual of the fiber class and where τ1[f ]= ∗ ∗ evs1 (f ) ∪ ψ1 is the corresponding ‘descendent constraint’ described at the end of Section 12. We will compute H in two different ways. The first is based on the standard method of ‘splitting the domain’, which yields the following general facts for 4-manifolds.

Lemma 15.1. Let X be a symplectic 4-manifold with canonical class K. (a) For A =0and g =1the GW invariant with a single constraint B ∈ H2(X) is 1 (15.6) GW (B)= K · B. 0,1 24

(b) For any classes A, f ∈ H2(X) satisfying A · K = −1, · ∗ (f A) 2 GWA,1(τ1[f ]) = (A + K · A)GWA,0 24 · · + (f A2)(A1 A2)GWA1,1GWA2,0.

A1+A2=A A1=0,A2=0

Proof. (a) For ν =0,M1,1(X, 0) is the space M1,1 × X of ‘ghost tori’ f :(T 2,j) → X with f(z)=p a constant map. At such f, the ﬁber of the obstruction bundle is H1(T 2,f∗TX)=H1(T 2, O) ⊗ TX. The dual of 1 2 the bundle H (T , O)overM1,1 is the Hodge bundle. Since the ﬁrst Chern number of the Hodge bundle is 1/24, the Euler class of the obstruction bundle is 1 χ(X)[M ] ⊗ 1+ 1 ⊗ K ∈ H (M × X). 1,1 24 2 1,1 THE SYMPLECTIC SUM FORMULA 1019

For ν = 0, the (virtual) moduli space is the zeros of a generic section of the obstruction bundle, which consists of (i) maps from a torus with any complex structure to χ(X) speciﬁed points of X and (ii) maps from a torus of speciﬁed complex structure into some point on the canonical divisor. Generically, the images of the type (i) maps will miss the constraint surface representing B. The maps of type (ii) give the formula (15.6). (b) The genus 1 topological recursive relation says

∗ 1 α GWA,1(τ[f ]) = GWA,0(Hα,H ,f) 24 α + GWA1,1(Hα)GWA2,0(H ,f)

A1+A2=A α α ∗ where {Hα} and {H } are bases of H (X) dual by the intersection form. α β α But for A =0GW A,0(H ,H ,f) vanishes by dimension count unless H and Hβ are two-dimensional, and then each A-curve hits a generic geometric α representative of Hα at H · A points counted with algebraic multiplicity. A dimension count also shows that the moduli spaces with A1 = A and A2 =0 are of the wrong dimension to contribute to the double sum above. Hence the expression above becomes 1 α (H · A)(Hα · A)(f · A)GWA,0 24 · α · · + (Hα A1)(H A2)(f A2)GWA1,1GWA2,0

A1+A2=A A1=0,A2=0 plus the term with A1 = 0, which by (15.6) is 1 1 (K · H )GW (Hα,f)= (K · H )(A · Hα)(A · f)GW . 24 α A,0 24 α A,0 α The lemma follows because (Hα · A1)(H · A2)=A1 · A2.

Taking X to be the rational elliptic surface E, we can apply Lemma 15.1 with A = s + df . Then K = −f, A · f = 1 and A2 =2d − 1. The only possible decompositions are A1 = kf and A2 = s +(d − k)f so that: d ∗ d − 1 GW (τ[f ]) = GW + k GW GW − . s+df , 1 12 s+df , 0 kf,1 s+(d k)f,0 k=1

But for the rational elliptic surface the invariant GWkf,1(s)isσ(k) for k>0 (since in P2 there is a unique cubic through 9 generic points). As in Section 4 of [IP1], for each k there are σ(k) distinct k-fold covers of an elliptic curve with a marked point, all with positive sign. Because the marked point can go to any of s · kf = k points, this means that the unconstrained invariant is

GWkf,1 = σ(k)/k for k>0. 1020 ELENY-NICOLETA IONEL AND THOMAS H. PARKER

It follows that 1 (15.7) H(t)= tF − F + F · G. 12 0 0 0 On the other hand, we can calculate H(t) by splitting the target and using the symplectic sum theorem. Let P = T 2 × S2, and let F denote both a fiber of the elliptic fibration E and a fixed torus T 2 ×{pt} inside P. We can apply the sum formula by writing E = E#F P for the class A = s + df with the constraint on the P side. Since A · F = 1, the connected curves representing A split into the union of connected curves in E and in P; thus the symplectic sum formula applies for the GW (as well as the GT invariants). If we have a genus 0 curve on the P side in the class s + d1F , then by projecting onto the T 2 factor and noting that there are no maps from S2 to 2 T of nonzero degree, we conclude that d1 = 0. But the moduli space of genus 0 curves in P representing s and passing through F is isomorphic to F = T 2, and moreover the relative cotangent bundle to them along F is isomorphic to the normal bundle to F . Thus, ∗ ∗ ∗ GWs,0(τ1[f ]) = GWs,0(f ∪ f )=0. We conclude that there is no contribution from genus 0 curves on the P side or in the neck (which is also a copy of P). The same argument shows that there are no rational curves in F , and so the g = 0 absolute and relative invariants are the same. With these observations, the only possibility is to have a genus 1 curve on the P side, genus 0 on the E side, and no contribution from the neck. The symplectic sum formula thus says ∗ · P ∗ GWd,1(τ1[f ]) = GWs+d1f,0(E) GWs+d2f,1( )(τ1[f ]).

d1+d2=d This last invariant can be computed by applying the topological recursive relation to X = P just as in Lemma 15.1: ∗ d − 1 GW (τ [f ]) = GW + d GW GW s+df , 1 1 12 s+df , 0 1 d1f,1 s+d2f,0 d1+d2=d d1=0,d2=0

+d2 GWs+d1f,1 GWd2f,0.

But the invariants of P satisfy GWdf , 0 =GWs+df , 0 = 0 for d = 0 by the projection argument above, while for d = 0, Lemma 14.4 gives d1GWd1f,1 = GW (s)=2σ(d ). We therefore get d1f,1 1 1 (15.8) H =2F · G − . 0 24

Combining (15.7) with (15.8) and noting that F0(0) = GWs,0 =1· ts we see that F0 satisﬁes the ODE · tF0 =12G F0 THE SYMPLECTIC SUM FORMULA 1021 with F0(0) = 1 · ts. Hence

F0(t)=ts exp 12 G(t)/t dt .

Using the Taylor series of log(1 − t) and some elementary combinatorics, we obtain 12 1 F (t)=t . 0 s 1 − td d It remains to show (15.4) for g>0. This case is different because for genus g>0 the relative invariants are no longer equal to the absolute invariants. We start by fixing a fiber F of E and introducing two generating functions for the genus g relative invariant: one recording the number of curves passing through g points in E \ F , the other recording the number of curves passing through g − 1 points in E \ F plus a fixed point on F : V F g d Fg (f)= GWs+df , g (p ; C1(f))t , d V F g−1 d Fg (p)= GWs+df , g (p ; C1(p))t . d Using Lemma 14.8, we can relate the absolute and relative g = 1 invariants of E.

Lemma 15.2. For X = E, the absolute and relative g =1invariants in the classes s + df ∈ H2(E(1)) are related by equations V V · (a) Fg = Fg (p)+Fg−1(f) G , V (b) Fg = Fg (f), V · · V (c) 0 = Fg (p) F0 + Fg−1 F1 (p).

2 2 Proof. To prove (a), we again write E = E#F P where P = T × S , and put g − 1 points on E and the remaining point on P. If we start with a class s + df , the only possible decompositions are s + af and s + bf where d = a + b. Since there are g − 1 points on the E side, the genus g1 ≥ g − 1. There are two possibilities: (1) Genus g in class s + df on E and genus 0 in class s + bf on P. But that forces b = 0 so that a = d.

(2) Genus g − 1 in class s + df on E and genus 1 in class s + bf on P. Putting these together gives (a). Relation (b) is a reformulation of Lemma 14.8. Relation (c) is seen by applying the symplectic sum formula to the sum K3=X1#F X2 where X1 = X2 = E (the elliptic surface K3=E(2) is the 1022 ELENY-NICOLETA IONEL AND THOMAS H. PARKER

ﬁber sum of E = E(1) with itself). Because a generic complex structure on K3 admits no holomorphic curves, then all relative and absolute invariants of K3 vanish. In particular, the genus g invariants through g − 1 points in the class [s + df ] ∈ H2(K3)/R vanish, where R is the set of rim tori corresponding to the gluing K3=E#F E. Now, for any g ≥ 1, put all the g − 1 points on X1 and split as above. A dimension count shows that the genus of the curve on X1 must be at least g − 1, so the only possible decompositions are:

(1) A genus g curve in the class s + d1F on X1 and a genus 0 curve on X2 in the class s + d2f, d = d1 + d2;

(2) A genus g − 1 curve in the class s + d1F on X1 and a genus 1 curve on X2 in the class s + d2f, d = d1 + d2.

V · V · V The symplectic sum formula then gives 0 = Fg (p) F0 + Fg−1(f) F1 (p), which simpliﬁes by (b).

Formula (15.4) follows quickly from Lemma 15.2. Taking g = 1 in Lemma V 15.2c and factoring out F0 = 0 yields F1 (p) = 0. Putting that in Lemma 15.2a V and again noting that F0 = 0, we see that Fg (p) = 0 for all g>0. Parts (a) and (b) of Lemma 15.2 then reduce to Fg = Fg−1 · tG which gives (15.4) by induction.

16. Appendix: Expansions of relative GT invariants

The Gromov-Witten invariants described in Section 1 are homology elements — the pushforward of the compactified moduli space under (1.18). These can be assembled into a power series (1.24) with coefficients in homology. Often, however, it is convenient to write the GW and GT invariants as power series whose coefficients are numbers, preferably numbers with clear geometric interpretations. This appendix describes how this can be done for the relative GT invariants which appear in the symplectic sum formula. Such series expansions are easiest when we can ignore the complications HV caused by the covering (1.20), replacing the space X,A,s by the more easily ∼ (s) understood space Vs = V . This can be done by pushing the homology class of the invariant down under the projection ε of (1.20), obtaining a ‘summed’ GW series V V V (A.1) GWX = ε∗ GWX = GWX,A tA

A∈H2(X) THE SYMPLECTIC SUM FORMULA 1023 whose coefficients are homology classes in sVs. This is a less refined invariant, but has the advantage that its coefficients become numbers after choosing a basis of H∗(V ). Of course (A.1) is the same as the original GW invariant when the set R of (1.19) vanishes, that is, when there are no rim tori. This occurs whenever H1(V ) = 0 or more generally when every rim torus represents zero in H2(X\V ). We will describe the numerical expansion under that assumption; the same V discussion applies in general to GWX . HV ∼ (s) When there are no rim tori, X,A is the union of those Vs = V with deg s = A · V . Fix a basis γi of H∗(V ; Q). Then a basis for the tensor algebra on N × H∗(V ) is given by elements of the form

(A.2) C = C ⊗···⊗C s,I s1,γi1 s,γi ≥ { ∨ } ∈ ∗ M where si 1 are integers. Let Cs,I denote the dual basis. When κ H ( ) and α ∈ T∗(X), we can expand 1 ∨ − (A.3) GTV (κ, α)= GTV (κ, α; C ) C t λ χ. X (s)! X,A,χ s,I s,I A s,I

The coefficients in (A.3) have a direct geometric interpretation. Choose generic pseudomanifolds K ⊂ Mg,n, Ai ⊂ X, and Γj ⊂ V representing the Poincaré duals of κ, α, and the γj in their respective spaces. Then V GTX,A,χ(κ, α; Cs,I ) is the oriented number of genus g (J, ν)-holomorphic, V - regular maps f : C → X with C ∈ K, f(pi) ∈ Ai and having a contact of order sj with V along Γj at points xj. Because of that interpretation, the Cs,I are called “contact constraints”. While for the analysis it is important to work with ordered sequences s, in applications it is more convenient to forget the ordering. The symmetries of the GW invariants allow us to replace the basis (A.2) with the one having elements of the form ma,i (A.4) Cm = (Ca,γi ) a,i where m =(ma,i) is a finite sequence of nonnegative integers. Generalizing (1.16), we write

(A.5) ma,i |m| = a ,m!= ma,i!,(m)= ma,i, deg m = a · ma,i. a,i a,i a,i a,i

Let {za,i} denote the dual basis; these generate a (super-)polynomial algebra with the relations za,i zb,j = ± zb,j za,i where the sign is + if and only if (deg γi)(deg γj) is even. Then the generating series of the relative GT invariant 1024 ELENY-NICOLETA IONEL AND THOMAS H. PARKER is

ma,i V V (za,i) −χ (A.6) GTX (κ, α)= GTX,A,χ(κ, α; Cm) tA λ ma,i! A,g m a,i where the sum is over all sequences m =(ma,i) as above and where the coeﬃ- V · cients GTX,A,χ(κ, α; Cm) vanish unless deg m = A V . Formally, this generating series (A.6) is given by    V V    −χ GTX (κ, α)= GTX,A,g κ, α; exp Ca,γi za,i tA λ . A,g a,i

University of Wisconsin, Madison, WI E-mail address: [email protected] Michigan State University, East Lansing, MI E-mail address: [email protected]

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